# 3.1: Assessments - Mathematics

### Homework Sets, Exercises, Quizzes, and Written Assignments for this course are all located in MyOpenMath.

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• Mathematics for the Liberal Arts I.

## Lesson 1

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Attribution: Hippo cartoon, by agomjo. Public Domain.. Source.

Diego took a picture of a hippo and then edited it. Which is the distorted image? How can you tell?

Is there anything about the pictures you could measure to test whether there’s been a distortion?

### 1.2: Sketching Stretching

A dilation with center (O) and positive scale factor (r) takes a point (P) along the ray (OP) to another point whose distance is (r) times farther away from (O) than (P) is. If (r) is less than 1 then the new point is really closer to (O) , not farther away.

Dilate (H) using (C) as the center and a scale factor of 3. (H) is 40 mm from (C) .

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### 1.3: Mini Me

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Description: <p>Figure on the right side with large circle A as body and small circle C as head. Segments H I and I J are left arm connect to circle A at point H and angle down. Segments E F and F G are right arm connect to circle A at point E and angle up. Point P is all the way to the left and has dashed lines to point D on circle C and to point B on circle A.</p>

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1. Dilate segment (AB) using center (P) by scale factor (frac12 ) . Label the result (A'B') .
2. Dilate the segment (AB) using center (Q) by scale factor (frac12) .
3. How does the length of (A''B'') compare to (A'B) ? How would the length of (A''B'') change if (Q) was infinitely far away? Explain or show your answer.

### Summary

A scale drawing of an object is a drawing in which all lengths in the drawing correspond to lengths in the object by the same scale. When we scale a figure we need to be sure to scale all of the parts equally or else the image will become distorted.

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor. The scale factor is the factor by which every length in a original figure is multiplied when you make a scaled copy. A scale factor greater than 1 enlarges an object while a scale factor less than 1 shrinks an object. What would a scale factor equal to 1 do?

For example, segment (BC) is a scaled copy of segment (DE) with a scale factor of (frac14) . So (BC=frac14DE) . If (DE=6) , then (BC=frac64) or 1.5.

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To perform a dilation, we need a center of dilation, a scale factor, and something to dilate. A dilation with center (A) and positive scale factor (k) takes a point (D) along the ray (AD) to another point whose distance is (k) times farther away from (A) than (D) is.

Segment (FG) is a dilation of segment (DE) using center (A) and a scale factor of 3. So (FA=3 oldcdot DA) . If (DA=15) , then (FA=45) .

## Introduction

### Previous research on classroom assessment

#### Effects of classroom assessment on student achievement

In educational research, often large positive effects of teachers’ use of classroom assessment on student achievement have been reported (studies reviewed in Black and Wiliam 1998, or more recently in Briggs et al. 2012, and in Kingston and Nash 2011). Notwithstanding the fact that scholars of these studies in most cases refer to classroom assessment or formative assessment when they discuss their research, the similarity of the operationalization they opted for is quite low many different definitions and assessment methods have been used under the same umbrella term of classroom assessment (see Veldhuis and Van den Heuvel-Panhuizen 2014). What strings studies on classroom assessment together, however—in addition to the terminology used—is that most interventions are focused on enhancing teachers’ subject knowledge and promoting the use of assessments, thus allowing teachers to subsequently provide formative feedback to students. Formative feedback means “information communicated to the learner that is intended to modify the learner’s thinking or behavior for the purpose of improving learning” (Shute 2007, p. 1). This type of feedback has been found to be most effective for motivating students and improving their learning (e.g., Hattie and Timperley 2007). In addition to the fact that the research projects on the effects of classroom assessment and their interventions were small scale, their comparability has been criticized because of the different conceptualizations of what classroom assessment entails (e.g., Bennett 2011). Even though the specificities of studies that have shown the effect of classroom assessment are different, their results do point to the effectiveness of the use of classroom assessment for improving students’ mathematics achievement. On the basis of such empirical results, recently, in the USA, the National Council of Teachers of Mathematics (NCTM 2013) strongly endorsed teachers using classroom assessment strategies in their daily instruction in mathematics education. The basic idea behind the effectiveness of teachers’ use of classroom assessment is that it can lead to teachers gaining more relevant and useful information on their students’ understandings and skills. This allows them to subsequently better adapt their teaching to their students’ needs, which in turn is expected to lead to improved student achievement. A recent study with 45 primary school teachers in Sweden (Andersson and Palm 2017) confirmed this line of reasoning with a yearlong intensive professional development program on using formative assessment strategies, resulting in students of these teachers significantly outperforming students in the control group on a mathematics posttest.

#### Strategies for using classroom assessment in mathematics

A number of scholars have focused on providing teachers with strategies for using classroom assessment in mathematics (e.g., Andersson and Palm 2017 Keeley and Tobey 2011 Leahy et al. 2005 Torrance and Pryor 2001 Wiliam 2011 Wiliam et al. 2004). These strategies for classroom assessment often concern activities that are familiar to teachers but that are now used with a clear assessment focus (e.g., Wiliam 2011). An example of such an assessment strategy is an all-students response system with multiple choice cards (ABCD cards) this means that the teacher poses a question that touches a key aspect of what is currently taught in class and to which all students respond individually by holding up a card. Teachers can use the information gathered in this way to go over a particular explanation or subject again, or instead move on an instructional decision teachers make on a day-to-day basis. Such activities can be seen as an operationalization of the framework Wiliam and Thompson (2007) proposed, consisting of five key strategies that make up teachers’ and students’ formative assessment practice. These strategies are aimed at assisting teachers and students in establishing the following three pieces of information about the learner: where (s)he is going, where (s)he is right now, and how to get there (see also Stiggins et al. 2004). In this framework, the how-to-get-there part consists of the teacher providing formative feedback that moves learners forward (Hattie and Timperley 2007). The other key strategies are related to sharing learning goals, making use of effective classroom discussions and learning tasks that elicit evidence of student understanding, and activating students as owners of, and resources for, their own learning (Wiliam and Thompson 2007, p. 63).

To investigate how teachers can acquire useful knowledge about their students’ learning,

a number of studies have investigated the influence of interventions focusing on supporting teachers in their assessment practice. For example, in Wiliam et al. 2004), science and mathematics teachers were supported in their assessment practice over the course of two school years. This resulted in large learning gains, but this study—as it was not an experimental study—was focused on determining principles for practice and not so much on establishing the effect of the support. Another example is the investigation of the influence of a long-term (4 years) professional development program on formative assessment design in biology education on teachers’ formative assessment abilities (Furtak et al. 2016). They found that teachers improved their assessment abilities on several aspects, such as question quality, interpretation of student ideas, and feedback quality, but surprisingly not on task quality meaning that they did not provide students with higher quality tasks. In this sense, the professional development does not always lead to the envisioned results. A similar result was found in a large-scale study in the USA (Randel et al. 2016) in which a widely used program on classroom assessment was experimentally evaluated, and neither students’ mathematics achievement nor teachers’ assessment practices appeared to have been influenced. Another example is a study on formative assessment in science education: despite training teachers in the use of formative assessment, their students’ achievement levels in science did not improve (Yin et al. 2008). These scholars hypothesized that this result was most probably due to a suboptimal implementation of the formative assessment strategies. A possible other reason could be that the aforementioned studies all aimed to assist teachers in their assessment by providing them with general strategies for formative assessment. This means that although they were meant to be used in the domain of mathematics (or science), there was no close relationship with the taught content when assessing the students. Instead, the focus was more on the format of the assessment techniques or the accompanying feedback. Formative assessment techniques that are closely connected to the taught mathematics have the potential to really inform teachers’ further instruction. Such a content-dependent approach was chosen by Phelan et al. (2012). In their study, teachers were supported to assess students’ learning in pre-algebra. To find out what had to be assessed, an expert panel was organized to map algebra knowledge and its prerequisites. This map was used to design the questions that could provide teachers with the necessary information, which turned out to have had a positive impact on students’ learning (Phelan et al. 2012).

#### Features of high-quality teacher professional development

To ensure that teachers optimally implement what they learn about classroom assessment, a number of features of high-quality teacher professional development have to be carefully considered (see e.g., Garet et al. 2001). Although the direct usefulness of such lists of features has been questioned (e.g., Beswick et al. 2016 Kennedy 2016), they do provide actionable ideas to assist shaping professional development programs. For example, as shown in the study by Phelan et al. (2012), it is important to focus on the mathematical content. However, as Kennedy’s (2016) findings suggest, a professional development program should not exclusively focus on content knowledge, but should also help teachers to expose student thinking. Classroom assessments should be linked to the learning trajectories, the standards, the curriculum, and the textbooks. In connection with this, the professional development should ensure that teachers are aware of the mathematics teaching and learning trajectories (or learning progressions) of the grades they teach (Bennett 2011). To achieve this knowledge, mathe-didactical analyses (Van den Heuvel-Panhuizen and Teppo 2007) of the mathematical content to be taught should be carried out leading to an in-depth knowing of what concepts and skills are important, how models and strategies are related, and how they evolve over the years. Without this knowledge, worthwhile assessments are impossible. Secondly, having teachers collectively discuss and reflect on student work or reactions can support teachers’ engagement in active learning (e.g., Lin 2006). Finally, the professional development should also take place over a prolonged period of time, because it often takes time for teachers to implement what they learned about classroom assessment (e.g., Black and Wiliam 1998) although changes have also been shown to occur rather rapidly (see Liljedahl 2010).

To establish a collaborative teacher learning community, gradualism, flexibility, choice, accountability, and support are deemed necessary (Wiliam 2007). Repeated meetings allow teachers to integrate what they have learned in their own practice and see their own practice in new ways, leading to new thinking. When teachers are also involved in the design of the classroom assessment, in collaboration with other teachers, it can raise their awareness of not only students’ mathematics learning difficulties but also their decisions about remedial instruction (Lin 2006). Furthermore, for teachers to use the information gathered through their use of assessment, they have to be involved in the process of development of the assessment, have an active role in how they use the assessment, and be able to use the assessment information (Wilson and Sloane 2000, p. 191). Recently, Heitink et al. (2016) described prerequisites for the implementation of classroom assessment (or what they call assessment for learning) in teachers’ classroom practice that reflect the previously mentioned features of effective professional development. Among the main prerequisites was that classroom assessment tasks should be meaningful and closely integrated into classroom instruction. Furthermore, teachers have to be able to interpret assessment information on the spot and the assessment should provide useful and constructive feedback that can be used for further instruction. Integrating these insights on professional development and the findings about the effectiveness of teachers’ use of classroom assessment led us to set up our current study.

### The present study

In the study described here, we focused on investigating the effect of supporting grade 3 teachers in the development and use of classroom assessment in their classrooms in mathematics on their students’ achievement. In giving this support, we strived to integrate the previously mentioned features of classroom assessment in mathematics education and effective professional development. In order to keep the classroom assessment closely connected to the taught mathematics, the assessment was based on mathe-didactical analyses of the important mathematical content that was at hand in the period of the school year when the support would be provided (see also Kim 2019). The mathematics textbook teachers used in their classrooms was the main resource for determining this content because, in the Netherlands, the textbook content can be considered the curriculum (e.g., Meelissen et al. 2012). In addition, to have the assessment tied with the teaching process, the use of classroom assessment was conceived as the use of classroom assessment techniques (CATs) short teacher-initiated assessment activities that reveal students’ understanding of a particular mathematical concept or skill. Our main research question was What effect does supporting third-grade teachers in the development and use of CATs in mathematics education have on their students’ mathematics achievement? How teachers use the different CATs could have an influence on the resulting effects on student achievement therefore, we also qualitatively explored how teachers use and implement CATs in their classroom practice.

## Level 3 Mathematics and statistics assessment resources

These resources are guides to effective assessment and should not be used as actual assessment.

These are publicly available resources so educational providers (including teachers and schools) must modify them to ensure that student work is authentic.

Teachers will need to set a different context or topic to be investigated identify different texts to read or perform or change figures, measurements or data sources to ensure that students are demonstrating that they can apply what they know and can do.

The Level 3 achievement standards for Mathematics and statistics are registered and have been published on the NZQA website.

Exemplars of student work or expected student responses (written by subject moderators) have been developed for Level 3 achievement standards. The exemplars are on the NZQA Subject Specific Resources pages on the NZQA website and are all available for use.

Assessment resources and exemplars for all Level 3 externally assessed standards have been published on the NZQA website.

You can take as many MPAs (math placement assessments) as you like. LUC (Loyola University Chicago) pays for the first bundle of five. Every subsequent bundle of five costs $15, payable online, through the ALEKS link in LOCUS. Each bundle consists of one unproctored, practice MPA, the score of which is not used to place you in a class, and four proctored MPAs, the scores of which are used to place you in a class and six months of access to Learning Modules (if initiated within six months of the first MPA). Any unused assessments and time in Leanring Modules expire on January 31 of each year. Before coming to LUC, take the unproctored, practice MPA as if it were proctored. That is, finish it in 90 minutes. Use only blank paper and a pencil or pen. Do not use any assistance of any kind, including but not limited to, calculators, search engines, math web sites, notes, books, or people. Once the first MPA is finished, the MPA (ALEKS) will generate Learning Modules based on your correct and incorrect answers specific to you. Use them to refresh/increase your math memory/knowledge. Use the Learning Modules for an interval of 30 to 45 uninterrupted minutes. Do that two or three times a day for several days. That should prepare you to do better on the next MPA. To take the unproctored practice MPA and to access the Learning Modules, log into the locus.luc.edu Please enter your LUC UVID (the letters appearing before "@luc.edu" in your LUC email address, your username) and your password. Within the LOCUS Student Homepage, select Student External System (the bottom-right option). In the navigation menu to the left, select ALEKS (the second option). To take the proctored MPA, the results of which are used to place you in a math class, see the next FAQ: When and where do I take a proctored MPA?. The following instructions are very detailed and contain a great deal of information regarding the MPA. Please read them completely and thoroughly. You must take a proctored MPA in order to use the results to place you in a math class. Because of COVID-19, all proctored MPAs last 180 minutes under the principle of universal design. Generally, students complete the MPA in 90 minutes. Use only blank paper and a pencil or pen. Do not use any assistance of any kind, including but not limited to calculators, cell phones, search engines, math web sites, cheat sheets, notes, books, or people. Each student has a unique knowledge state that consists of the math topics the student knows, does not know, and is ready to learn. The ALEKS placement assessment determines the unique knowledge state of each student. The system can then properly place students into a course based on their math abilities. Loyola University Chicago takes academic integrity very seriously and investigates all allegations of academic dishonesty. Integrity is a character-driven commitment to honesty, doing what is right, and guiding others to do what is right. Loyola University Chicago students and faculty are expected to act with integrity in their educational pursuits. Cheating on an academic evaluation or assignment is a violation of the academic integrity policy. If you are found to be in violation of the academic integrity policy, academic sanctions will be imposed. For more information, please refer to the academic integrity policy found here: https://www.luc.edu/academics/catalog/undergrad/reg_academicintegrity.shtml. Between each MPA there is a 24-hour waiting period and, concomitantly, a 3-hour minimum work requirement in the Learning Modules. Once those conditions are met, you can take the MPA at any time anywhere. A proctored assessment requires that a LockDown Browser be installed on your computer. ALEKS will prompt you to download and install the LockDown browser once. Once it is installed you do not have to do it again. ALEKS will initiate and run the LockDown browser. You will be recorded while taking the assessment. Your computer will need a video camera. ALEKS will prompt you through a process testing your video camera. It is highly recommended you use a desktop or laptop running Windows or Macintosh. Chromebooks cannot run the MPA. The ability to run the MPA on any device other than a Windows or Mac desktop or laptop is unknown. Items needed to take the assessment: • Pencil/pen and paper (to work out answers). Students may not use calculators or have any assistance of any kind except pencil and paper while taking the MPA. • Your LUC user name and password. • A photo ID. A high school or college ID is preferred. If it is a driver's license, cover up sensitive information: license number, date of birth, etc. • Computer with a webcam (You may wish to check your lockdown browser set-up prior to the second or “true” proctored MPA. See https://www.aleks.com/support/lockdown_test for instructions) The MPA requires the use of LockDown Browser and a webcam. The webcam can be built into your computer or can be the type that plugs in with a USB cable. The LockDown Browser will prevent you from accessing other websites or applications you will be unable to exit the test until all questions are completed and submitted. The Webcam ensures that you take the exam without assistance of any kind except a pencil and blank paper. • Windows: 10, 8, 7 • Mac: OS X 10.10 or higher • Web camera (internal or external) & microphone • A broadband internet connection, a wired connection is preferred. Watch this short video to get a basic understanding of LockDown Browser and the webcam feature. https://www.youtube.com/embed/XuX8WoeAycs?rel=0&ampautoplay=1 or https://web.respondus.com/student-help/ Click on Video: Overview of LockDown Browser When taking the proctored MPA that requires LockDown Browser and a webcam, remember the following guidelines: • Ensure you're in a location where you won't be interrupted • Turn off all other devices (e.g. tablets, phones, second computers) and place them outside of your reach • Close all programs on your computer except your browser that opens LOCUS. • Clear your desk of all external materials — everything but blank paper, a pencil or pen, keyboard and mouse. • Make sure you've allotted about 90 minutes to complete the MPA • Remain at your computer for the duration of the test To produce a good webcam video, do the following: o Do not wear baseball caps or hats with brims o Ensure your computer is on a firm surface (a desk or table). Do NOT have the computer on your lap, a bed, or other surface where the device (or you) are likely to move o If using a built-in webcam, avoid tilting the screen after the webcam setup is complete o Take the exam in a well-lit room and avoid backlighting, such as sitting with your back to a window If you need technical assistance with the MPA, please contact ALEKS Support at (https://mhedu.force.com/aleks/s/alekscontactsupport) The proctored assessment is password protected. The password is LUC2021 If you have questions or concerns about the MPA, please contact the Math Placement Coordinator, John Houlihan, [email protected] The current MPA is ALEKS PPL ("ALEKS"), Assessment and LEarning in Knowledge Spaces, Placement Preparation and Learning. ALEKS is a Web-based, artificially intelligent assessment and learning system, which uses adaptive questioning to quickly and accurately determine exactly what a student knows and doesn’t know. ALEKS then provides the student individualized topics (Learning Modules) which the student can use to learn the material with which the student had difficulty. When using the Learning Modules, ALEKS also provides the advantages of one-on-one instruction, 24/7, from virtually any Web-based computer for a fraction of the cost of a human tutor. Please be aware that the Learning Modules will be available to you for at most six months, if started within six months of your first MPA, but ending on January 31. The MPA assesses eleven major math topics. 1. Whole Numbers, Fractions, and Decimals 2. Percents, Proportions, and Geometry 3. Signed Numbers, Linear Equations and Inequalities 4. Lines and Systems of Linear Equations 5. Relations and Functions 6. Integer Exponents and Factoring 7. Quadratic and Polynomial Functions 8. Rational Expressions and Functions 9. Radicals and Rational Exponents 10. Exponentials and Logarithms See a complete detailed list here. Only students whose ACT/SAT math scores have not placed them into the math course they want or need (see http://luc.edu/math/placement/ ) and who have a link to ALEKS in LOCUS need to take the MPA. If your ACT math score was 28 or above, do not take the MPA. If your SAT-R math score was 660 or above, do not take the MPA. Students with majors listed here need to take math courses. Generally, in order to graduate in four years, Chemistry and pre-health majors need to start in MATH 118, Precalculus II while students majoring in Physics, Engineering, and/or Math & Statistics need to start in MATH 161, Calculus I. It is also strongly recommended that all transfer students who have appropriate transfer credit take the MPA if they will need additional math courses at LUC. If the score on the MPA indicates a lack of preparedness for the course, the student should work to mastering the topics in the ALEKS Learning Module and take the MPA until they have the appropriate score. New students should contact Undergraduate Admissions prior to May 1. After May 1, you may contact [email protected] to confirm that your new major requires math, and request access to the MPA. If you are registering prior to the date when the AP Calculus results are known, and your ACT and/or SAT math subscores do not place you into the math course you need/want (see luc.edu/math/placement), you should take the MPA. Click here to see the AP scores that qualify for credit. No. Loyola's math placement is specifically designed to place students into a Loyola University Chicago math course, or appropriate course requiring math pre-requisites. Possibly. Some courses require certain levels of math before you can take them. To find the prerequisites for a course, use the class search or course catalog features in LOCUS (if you don’t have a login, you can sign in as a guest) and look at the course descriptions. Contact an academic advisor. Your advisor will be listed in LOCUS on your Student Center. Incoming first year students should contact First and Second Year Advising. • By phone: 773.508.7714 • Online: http://www.luc.edu/fsya/aboutus/contactus/ • Email: [email protected] • In person: Sullivan Center for Student Services, Suite 260, 6339 N. Sheridan Ave. Chicago, IL 60660 Transfer, junior, or senior students should contact their College or School advisors. The proctored MPA must be completed within 90 minutes. For those who need extended time accommodations due to a documented disability, please contact Services for Students with Disabilities at least one week in advance of the scheduled MPA date. Additional information about utilizing accommodations for the MPA can be found here: https://www.luc.edu/sac/placementtests/ Every bundle of MPAs includes one (1) practice unproctored MPA which is not used for placement, four (4) proctored MPAs, which are used for placement, and six months of access to individualized Learning Modules. LUC pays for the first bundle. Each successive bundle is$15, payable online by the student.

Date of Implementation: 09/01/98
Date of Last Review: 11/14/18

Date & Subject of Amendments:

11/14/18 – Amended to reflect that multiple assessment instruments (plural) are used in our system. The new writing and formatting styles were also applied to the policy.

05/17/06 - Amended the title, replacing College Readiness with Course Placement. Removed all previous language and added a new Part 1 and Part 2. Also repeals Carry Forward SU Policy 4.9 Presentation of ACT Scores for Enrollment in a Minnesota State University.

## 3.1: Assessments - Mathematics

At the end of each unit is the end-of-unit assessment. These assessments have a specific length and breadth, with problem types that are intended to gauge students' understanding of the key concepts of the unit while also preparing students for new-generation standardized exams. Problem types include multiple-choice, multiple response, short answer, restricted constructed response, and extended response. Problems vary in difficulty and depth of knowledge.

Teachers may choose to grade these assessments in a standardized fashion, but may also choose to grade more formatively by asking students to show and explain their work on all problems. Teachers may also decide to make changes to the provided assessments to better suit their needs. If making changes, teachers are encouraged to keep the format of problem types provided, which helps students know what to expect and ensures each assessment will take approximately the same amount of time.

In longer units, a mid-unit assessment is also available. This assessment has the same form and structure as an end-of-unit assessment. In longer units, the end-of-unit assessment will include the breadth of all content for the full unit, with emphasis on the content from the second half of the unit.

All summative assessment problems include a complete solution and standard alignment. Multiple-choice and multiple response problems often include a reason for each potential error a student might make. Restricted constructed response and extended response items include a rubric.

Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.

A note about technology use on assessments: Some assessments require use of technology, some allow it, and some prohibit it. These affordances or restrictions are communicated in each assessment narrative and in instructions to students. Reasons we chose to prohibit use of technology on some assessments include assessing a standard that requires students to sketch a graph by hand, and assessing a standard that requires students to use mathematical properties to rewrite expressions. Conversely, some standards specify that students must use technology for certain things, like generating a best-fit line and correlation coefficient. On assessments where these skills are assessed, technology is required. This approach is in keeping with many state and national standardized assessments that include calculator-allowed and calculator-prohibited portions. Our approach, though, is to allow or prohibit technology on an entire assessment—no single assessment in this curriculum contains both technology-allowed and technology-prohibited portions.

### Design Principles for Summative Assessments

Students should get the correct answer on assessment problems for the right reasons, and get incorrect answers for the right reasons. To help with this, our assessment problems are targeted and short, use consistent, positive wording, and have clear, undebatable correct responses.

In multiple-choice problems, distractors are common errors and misconceptions directly relating to what is being assessed, since problems are intended to test whether the student has proficiency on a specific skill. The distractors serve as a diagnostic, giving teachers the chance to quickly see which of the most common errors are being made. There are no trick questions, and the phrases "all of the above" and "none of the above" are never used, since they do not give useful information about the methods a student used.

Multiple response prompts always include the phrase "select all" to clearly indicate their type. Each part of a multiple response problem addresses a different piece of the same overall skill, again serving as a diagnostic for teachers to understand which common errors students are making.

Short answer, restricted constructed response, and extended response problems are careful to avoid compounding errors, where a part of the problem asks for students to use correct work from a previous part. This choice is made to ensure that students have all possible opportunities to show proficiency on assessments.

When possible, extended response problems provide multiple ways for students to demonstrate understanding of the content being assessed, through some combination of arithmetic or algebra, use of representations (tables, graphs, diagrams, expressions, and equations) and explanation.

### Rubrics for Evaluating Student Answers

Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses.

Restricted Constructed Response

• Tier 1 response: Work is complete and correct.
• Tier 2 response: Work shows general conceptual understanding and mastery, with some errors.
• Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 responsecan be given as the reason for a Tier 3 response instead of listing combinations.
• Tier 1 response: Work is complete and correct, with complete explanation or justification.
• Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
• Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors.
• Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.

Typically, sample errors are included. Acceptable errors can be listed at any Tier (as an additional bullet point), notably Tier 1, to specify exclusions.

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

• 20 questions, multiple choice
• The test software will select questions based on your previous answers. Answer all the questions to the best of your ability and do not skip any, as you cannot go back.
• Some questions provide access to a calculator the calculator appears as part of the question. Other questions will not allow access to a calculator.

The Preparation Guide is a reference tool for George Brown College students preparing to take a placement math assessment. The guides focuses on foundation-level math skills. The preparation guide does not cover all topics on the assessments. The Assessment Centre at George Brown College is not responsible for students' assessment results.

### Sample Questions

See pages 2-4. The answer key is on pages 5-8.

Accuplacer has created an App containing a sample Algebra (Quantitative Reasoning, Algebra, and Statistics) test. To find the sample test and review test content, please create an account.

## Perseverance and Mathematics - A mathematical journey to Mars

The landing of the Perseverance rover on Mars is an excellent catalyst for a discussion with students about the nature of Mathematics. It is a chance to inspire curiosity and wonderment and to do so through a mathematical lens. The video of a small, fragile craft descending towards the surface of an alien planet, immediately captures the imagination. As the lander bobs gently beneath its parachutes and as the NASA team communicate critical moments in its descent, one is filled with a sense of awe at what is possible when we dream big and dare mighty things. By the time the lander has fired its rockets to enter a hover above the surface, and as the rover is delicately lowered to the ground on slender cables, the viewer’s mind should be awash with questions. Now is the time to invite our students to ponder the beauty and wonder revealed in the mathematics of what they are observing.

Mathematics rightly viewed possesses not only truth but supreme beauty. - Bertrand Russell (1907) The Study of Mathematics

We often overlook the mathematics that surrounds and makes so much of our daily lives possible. It is easy to think of mathematics as something that belongs inside the walls of a particular type of classroom. It is equally easy to think that mathematics is about the mere mechanics of solving a problem quickly and accurately. Contemplating the nature of the mathematics involved in landing Perseverance on Mars encourages a fresh perspective on a discipline many feel we escaped when we finished school. The Perseverance landing is awash with mathematics, and it should be readily apparent that it is mathematics that makes the whole endeavour possible. This is the mathematics of the real world, where formidable challenges are confronted with questions and creativity.

“Mathematics is not about numbers, but about life. It is about the world in which we live. It is about ideas. And far from being dull and sterile as it is so often portrayed, it is full of creativity”. - Keith Devlin - 2001

I had the pleasure of sharing the video of the landing with my class of Year Five students. As they watched, I invited them to notice and name the mathematics that might have been required at varying stages throughout the landing. The conversation was enlightening and evolved over time as fresh ideas emerged. From an initial focus on altitude and speed, students began to contemplate the wider range of factors that would need to be considered and included in calculations. The mass of the rover, the rate of deceleration, the gravity of Mars, the size of the parachutes, the time delay from Earth, the speed of light, the orientation of the rover in 3D space, the challenges of navigating a distant planet. We moved well beyond the traditional highly siloed approach where concepts (or curriculum outcomes) are examined in splendid isolation. Here the mathematics was meaningful, complicated, integrated and beautiful.

I’m simply stunned when I think how rarely formal learning gives us a chance to learn the whole game from early on. When I and my buddies studied basic arithmetic, we had no real idea what the whole game of mathematics was about. . . . It was kind of like batting practise without knowing the whole game. Why would anyone want to do that? - Perkins, David. (2009) Making Learning Whole

Investigating the mathematics behind Perseverance’s landing challenges our traditional approach to the discipline. School mathematics sadly has little in common with real mathematics.

Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that it is an aesthetic, creative, and beautiful subject. - Jo Boaler - (2015) Mathematical Mindsets

And, we know that our emphasis on the rote learning of mathematical processes is not facilitating the sort of deep-understanding of mathematics that our students need for success. Research from the Office of Australia’s Chief Scientist examined the approach taken to mathematics in 619 Australian schools achieving outstanding improvement in NAPLAN (a national standardised numeracy and literacy assessment) numeracy scores over a two-year period. A significant finding from this study was that “87% of case study schools had a classroom focus on mastery (i.e. developing conceptual understanding) rather than just procedural fluency.”

By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. - Paul Lockhart (2009) A Mathematician’s Lament

Hidden in Perseverance’s parachute is a beautiful example of the playful creativity of NASA’s mathematicians. Amidst the pattern designed to help NASA’s tracking systems monitor the performance of the parachute is a message only revealed by translating the visible pattern of wide and narrow segments of colour on the parachute into binary and then ASCII code. The message reads “Dare Mighty Things”. Maybe this is a message to teachers of mathematics and their students. A call for us to all dare mighty things as we explore a mathematics that is full of wonder and creativity.

## Abstract

Although formative assessment is regarded as a promising way to improve teaching and learning, there is considerable need for research on precisely how it influences student learning. In this study we developed and implemented a formative assessment intervention for mathematics instruction and investigated whether it had effects on students' interest and achievement directly and via students' perception of the usefulness of the feedback and their self-efficacy. We conducted a cluster randomized field trial with pretest and posttest. The 26 participating classes were randomly assigned to a control group or the intervention group. Results of path analyses indicate that feedback was perceived as more useful in the formative assessment condition, self-efficacy was greater, and interest tended to increase learning progress did not differ between the groups. The assumed indirect effects were partly confirmed: formative assessment showed an indirect effect on interest via its perceived usefulness.