By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
Mathematical Jargon Mathematical language is hard for beginners. Words such as 'or' and 'only' have more precise meanings than in everyday speech. Also confusing to beginners are words such as 'open' and 'field' that have been given specific mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon. Mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as rigor. Rigor Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based upon fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time; the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue amongst themselves about computer-assisted proofs. Because large computations are hard to verify, such proofs may not be sufficiently rigorous. Axioms in traditional thought were "self-evident truths,' but that conception is problematic. At a formal level, an axiom is just a string of symbols which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and thus, a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. Numerals and Naming Systems Some of the systems for representing numbers in previous and present cultures are well known. Roman numerals use a few letters of the alphabet to represent numbers up to the thousands, but are not intended for arbitrarily large numbers and can only represent positive integers. Arabic numerals are a family of systems originating in India, passing to medieval Islamic civilization and then to Europe, and now are the standard in global culture. They have undergone many curious changes with time and geography, but can represent arbitrarily large numbers and have been adapted to negative numbers, fractions, and other real numbers. Less-well-known systems include some that are written and can be read today, such as the Hebrew and Greek method of using the letters of the alphabet, in order, for digits 1–9, tens 10–90, and hundreds 100–900. A completely different system is that of the quipu, which the Inca used to record numbers on knotted strings. Finger Counting Many systems of finger counting have been, and still are, used in various parts of the world. Most are not as obvious as holding up a number of fingers. The position of fingers may be most important. One continuing use for finger counting is for people who speak different languages to communicate prices in the marketplace. Cognitive Theorists and Constructivists Constructivists believe that students may construct knowledge by themselves. In other words, students are actively engaged in the construction of their own knowledge. Students will assimilate and accommodate in order to build new knowledge, based on previous knowledge. Thus, in planning instruction based on constructivism, a teacher would focus on grouping designs, environment, problem-solving tasks, and inclusion of multiple representations. The goal in such a classroom would be for students to construct knowledge on their own. There are different levels of constructivism, including weak constructivism and radical constructivism. Cognitivists differ from constructivists in that they believe that active exploration is important in helping students make sense of observations and experiences. However, the students are not expected to invent or construct knowledge by themselves. They are only expected to make sense of the mathematics. In planning instruction based on cognitivism, a teacher would employ similar methods to those discussed above, with the focus on active exploration. Students would do a lot of comparisons of mathematical methods in making sense of ideas. Constructivism Three types of constructivism are weak constructivism, social constructivism, and radical constructivism. Weak constructivists believe that students construct their own knowledge, but also accept certain preconceived notions or facts. Social constructivists believe that students construct knowledge by interacting with one another and holding discussions and conversations. Radical constructivists believe that all interpretations of knowledge are subjective, based on the individual learner. In other words, there is no real truth; it is all subjective. Classroom instructional planning based on a weak constructivist viewpoint might involve incorporation of some accepted theorems and definitions, while continuing to plan active explorations and discussions. Planning based on a social constructivist viewpoint might involve group activities, debates, discussion forums, etc. Planning based on a radical constructivist viewpoint would involve activities that are open-ended, where there is more than one correct answer. The problems would invite more than one correct answer. Project-Based Learning Project-based learning is learning that centers on the solving of a problem. Students learn many different ideas by solving one 'big' problem. For example, for a unit on sine and cosine functions, a teacher may design a problem whereby the students are asked to model a real-world phenomenon using both types of functions. Students must investigate the effects of changes in amplitude, period, shifts, etc., on the graphs of the functions. Students will also be able to make connections between the types of functions when modeling the same phenomenon. Such a problem will induce high-level thinking. Project-based learning is derived from constructivist theory, which contends that students learn by doing and constructing their own knowledge. Cooperative Learning Cooperative learning simply means that students will learn by cooperating with one another. Students will be placed into groups of a size determined by the teacher. With such an approach, students work together to succeed in learning. Students may work together to learn a topic, complete an assignment, or compete with other groups. Examples of cooperative learning include Think-Pair-Share and Jigsaw.
Think-Pair-Share is a cooperative learning strategy that involves thinking about some given topic, sharing ideas, thoughts, or questions with a partner, and then sharing the partner discussion with the whole group. For example, in the mathematics classroom, a teacher may ask the class to think about the meaning of a proportional relationship. Each student would think for a set period of time, share ideas with a partner, and then each partner group would share their ideas regarding the meaning of proportionality.
Jigsaw is another cooperative learning strategy that involves dividing among each group member reading material or ideas to be learned. Each student will then read his or her information, summarize it, and share the findings or ideas with the group. In mathematics, students might be given information on modeling with cosine and sine functions. Students could then share what they learned about real-world phenomena modeled by each. Different students may also be assigned to read in-depth material on amplitude, period, shifts, etc. Control Strategies 'Control strategies' is another name for 'metacognitive learning strategies,' which indicate any strategy that promotes a learner's awareness of his or her level of learning. With such strategies, the student will work to determine what he or she knows and does not know regarding a subject. Possible control strategies are thinking, self-regulation, and discussing ideas with peers. Example: A student may discover his or her level of 'knowing' about functions by keeping a journal of any questions he or she might have regarding the topic. The student may list everything that he or she understands, as well as aspects not understood. As the student progresses through the course, he or she may go back and reconfirm any correct knowledge and monitor progress on any previous misconceptions. Memorization and Elaboration Strategies Memorization is simply a technique whereby rote repetition is used to learn information. Elaboration strategies involve the connection of new information to some previously learned information. In mathematics, for example, students may use elaboration strategies when learning how to calculate the volume of a cone, based on their understood approach for calculating the volume of a cylinder. The student would be making connections in his or her mind between this new skill and other previously acquired skills. A memorization technique would simply involve memorization of the volume of a cone formula, as well as ways to evaluate the formula. Prior Knowledge Three ways of activating students' prior knowledge are concept mapping, visual imagery, and comparing and contrasting. With concept mapping, a student would detail and connect all known aspects of a mathematics topic. Ideas would be grouped into subgroups. Such an approach would allow a student to see what he or she does not know, prompting the activation of any prior knowledge on the subject. Visual imagery is simply the use of any pictures or diagrams to promote activation of prior knowledge. For example, giving a picture of Pascal's triangle would likely activate students' prior knowledge regarding the binomial theorem. Comparing and contrasting means that the student will compare and contrast ideas or approaches. For example, a student might be given a mapping of an inverse function. He or she could then compare and contrast this mapping to a known mapping of a function, in order to decide how they are the same and different. This would activate a student's prior understanding of functions and the definition thereof. Three methods for ascertaining, or assessing, students' prior knowledge are portfolios, pre-tests, and self-inventories. Portfolios are simply a compilation of prior student activity related to mathematics topics. For example, a portfolio might show a student's work with transforming functions. Pre-tests are designed to measure a student's understanding of mathematics topics that will be taught in the course during the year. Self-inventories are just what the name implies: inventories that ask the students to name, list, describe, and explain information understood about various mathematics topics. Once a teacher has assessed students' level of prior knowledge regarding some mathematics topic, he or she may use that information to scaffold the instruction. In other words, the teacher may decide to further break down the mathematics material into more integral parts. Exact processes or steps may be shown, including justification for using certain properties or theorems. More examples may be shown, while including examples of many different variations of problems, in order to ensure that students are not simply memorizing one approach that will be incorrectly applied to any problem of that sort. The teacher may also decide that more group work, peer cooperation, and discussion are needed. For example, suppose a teacher determines that students have very little understanding of logic and valid arguments. The teacher may decide to re-teach the creation of truth tables, including truth values for intersections and 'if p, then q' statements. The teacher may also decide to re-teach how a truth table may be used to show if an argument is valid. Students may be placed into groups and asked to determine the validity of several simple arguments. Once students understand the concept, they may move on to more rigorous arguments, including equivalence relations. Concept Whereby Usage of Manipulatives Would Increase Conceptual Understanding Understanding of how to solve one-variable equations would certainly be enhanced by using rods and counters. With this manipulative, the rod would represent the variable, or x, while the counters would represent the constants on each side of the equation. A sample diagram of the equation, , is shown below. Note that the vertical line represents the equals sign. In order to solve the equation (and isolate x), four counters may be removed from each side of the mat. This process is shown below: Now, the final illustration is: Thus, the solution is x = 4.
The manipulative helps students understand the meaning of the subtraction property of equality in action, without simply memorizing its meaning. Problems Problem #1 Explain how an understanding of the area under the normal curve may be supported by using a graphing calculator. Provide a sample problem and describe the steps involved in solving, using both a manual approach and a technological approach. The area under the normal curve may be found by calculating z-scores for certain endpoint values. The mean to z areas for these z-scores may be used to find the area. A graphing calculator may use the normalpdf function and ShadeNorm function in order to show the same area under the normal curve, between two values. Consider the following problem: The class average on a statistics exam is 90, with a standard deviation of 4 points. Find the percentage of students who scored above 87 on the exam. This problem may be solved manually by first calculating the z-score. Since the score falls below the mean, the area above the score will equal the sum of 0.5 (or the area of one-half of the normal curve) and the mean to z area, which is 0.2734. Thus, the area above the z-score of −0.75 is 0.7734. The percentage of students who scored above 87 was 77.34%. This problem may also be solved by using the graphing calculator: Enter normalpdf(x, 0, 1) into the y = screen. (This represents the normal curve having a mean of 0 and standard deviation of 1.) Choose 2nd Vars, ShadeNorm. Enter ShadeNorm(87,100,90,4). (This represents the lower bound, upper bound, mean, and standard deviation.) Record the area of approximately 0.77. Thus, the calculator also shows that approximately 77% of the students scored above an 87. Problem #2 Describe how the understanding of derivative and anti-derivative may be enhanced/supported using a graphing calculator. Provide an example and show how to solve, using a manual approach and technological approach. The derivative of an expression is the slope of a tangent line to the curve, at a specific point. The anti-derivative of an expression is the inverse operation of the derivative. Taking the derivative of the anti-derivative will give the original expression. The derivative and anti-derivative can be calculated manually as shown below: Given , the derivative is .
The anti-derivative is .
Thus, evaluation of the derivative and anti-derivative for an x-value of 2 gives and , respectively.
The student can confirm his or her derivative and anti-derivative expressions by evaluating the graphed functions for the same x-value. If the expressions were correctly determined, then evaluation of the derivative and anti-derivative for the x-value should give the same y-value, for each. Using the graphing calculator, the derivative and anti-derivative for a given point may be evaluated by entering the expression into the y = screen, graphing the function, selecting 2nd Trace, and then choosing dy/dx and . After selecting the derivative or anti-derivative, the x-value may be typed. Evaluation of the derivative or anti-derivative for that x-value will appear on the screen. Piaget's Cognitive Development Theory Piaget's cognitive development theory is aligned with constructivism. In fact, constructivism is built on his ideas. Piaget's cognitive development theory indicates that students actively participate in the construction of their own knowledge via assimilation and accommodation. Current cognitive theorists do not believe that students have to construct their own knowledge, but instead that they only have to make sense of what they are observing. The four stages of learning, as developed by Piaget, are sensorimotor, preoperational, concrete operational, and formal operational. The defined stages show the progression from concrete thinking to abstract thinking. In other words, a child would need an object to understand properties, in the first stage. By the fourth stage, the child would be able to think abstractly, without some concrete form. In mathematics, this idea might be illustrated by first working with diagrams and manipulatives of numbers and then later writing symbolic forms of the numbers, including the numerals. This would illustrate the progression from 0 to 7 years. In the years of 11 to adulthood, much deeper abstraction is utilized. For example, people would be able to discuss functions and general properties, without looking at any concrete graphs or representations. Progression That a Student Undergoes as He or She Learns Mathematics When learning mathematics, students begin with concrete representations and ideas. Later, students are able to abstract meaning and make generalizations. Students will also be asked to apply abstract ideas from one topic to another mathematics topic. In other words, students would move from concrete representations, ideas, and facts to symbolic representations and generalizations. Piaget outlined such a progression in his general four stages of cognitive learning. For example, a student may first learn about solving equations by using a balance scale. After the student understands the process, he or she can solve alone, using the symbolic equations. He or she would also be able to describe the process for solving any equation. Direct Instruction Versus Student-Centered Instruction Direct instruction is instruction whereby the teacher delivers all content knowledge to be learned, and students, more or less, passively listen. The teacher employs a step-by-step instruction method for learning content. Student-centered instruction is learning whereby the teacher serves as a facilitator of learning and students actively participate in their own learning. Research has shown that students show a higher level of procedural and conceptual understanding when learning in a student-centered approach. Direct instruction might be more appropriate when teaching basic or fundamental theorems. Student-centered learning might be more appropriate when helping students make connections or develop higher-level thinking regarding a topic. Cooperative Learning Task Versus Traditional Task Think-Pair-Share is an activity whereby a topic is first given for consideration on an individual basis. Next, the students are arranged in pairs and asked to discuss the topic (e.g., any questions, comments, generalizations, etc.). Finally, each pair will contribute to a whole-class discussion on the topic. In mathematics, students would likely develop a higher level of understanding by using such an activity as Think-Pair-Share when learning about trigonometric functions. For example, students might be asked to consider different real-world situations that may be modeled with sine and cosine functions. Students could individually make a list and then share with a partner. Each partner group could then contribute to a whole class list. This list could be used as a reference sheet. Implementing Technology in Classroom Instruction Technology may be implemented in the mathematics classroom in many ways. For example, Excel may be used to perform regressions, calculate lines of best fit, calculate correlation coefficients, plot residuals, show convergence or divergence of a sequence, etc. Calculators may be used to evaluate and graph functions, find area under the normal curve, calculate combinations and permutations, perform statistical tests, etc. Graphing software, such as GeoGebra, may be used to graph and explore many shapes and functions. Students may also use it to graph reflections, rotations, translations, and dilations. Modifying Instruction to Accommodate English-Language Learners In mathematics specifically, instruction may be modified to include illustrations of ideas, in addition to given words. Audio may also be included for problem tasks. English-language learners may also be grouped with other fluent English-speaking students in order to assist with learning of the mathematics topic. Students will be able to hear the conversation, in addition to seeing the topic in print. In addition, problems may be broken down into smaller pieces, which can help the student focus on one step at a time. Further, additional one-on-one time with the teacher may be needed, whereby the teacher reads aloud and illustrates examples to be learned. Effective Learning Environment for ELL Students Characteristics of an effective learning environment for ELL students include creation of a low threshold for anxiety, use of pictures to teach vocabulary and mathematics ideas, implementation of graphic organizers, explicit teaching of vocabulary words, and use of active learning strategies. The latter two are extremely important, since ELL students need to learn exact terms and exact definitions while also engaging with fellow students, as opposed to sitting alone at a desk. Research completed by professors at the University of Houston and University of California list collaborative learning, use of multiple representations, and technology integration as important facets of an effective learning environment for ELL students (Waxman & Tellez, 2002). Mathematics Question That is Closed-Ended and Then Rewritten in an Open-Ended Manner Closed-Ended: Look at the graph of . Decide if the graph represents a function. Open-Ended: Provide an example of an equation that represents a function. Provide an example of an equation that does not represent a function. Explain how the graphs of the two equations compare to one another. The first question will elicit a simple, straightforward response, or 'Yes, it is a function.' The second question prompts the student to come up with two equations and then describe how the graphs of the two equations would compare. There is more than one possible answer, and the student has to make a comparison as well. Good Questioning Response Techniques A few good questioning response techniques are:
The key to good questioning response techniques is to show the student that his or her comments are important and to connect those comments to other student comments. The student should feel that he or she has made a contribution to the community of learners. A teacher should always ask a meaningful, thought-provoking question and provide sufficient time for the student to provide a meaningful and well-thought-out response. Student answers should lead to more questions and ideas and not serve as an endpoint. NCTM Categories of Questions That Teachers Should Ask The professional standards describe five categories of questions that teachers should ask. These categories are:
1) working together to make sense of problems; 2) individually making sense of problems; 3) reasoning about mathematics; 4) providing conjectures about mathematics; and 5) making connections within and outside of mathematics.
Sample questions include 'What does this mean?,' 'How can you prove that?,' and 'What does this relate to?'. Categories 4 and 5 are high level and include questions that prompt students to invent ideas and make meaningful connections. Accountants and Mathematical Modeling Accountants use mathematical modeling in a variety of ways. For example, an accountant models the future value of a certificate of deposit (CD) using the compound interest formula. An accountant also may fit a regression line to a client's overall savings over x years. An accountant may model tax payments with residual plots. Accountants may use past income tax returns to predict future tax expenses. Accountants may compare rates of return when investing in different mutual funds, by fitting and comparing regression lines. Scientists and Functions Scientists use functions to model real-world phenomena. For example, scientists use quadratic functions to model the height of an object tossed into the air or dropped from a certain height. Scientists use sine and cosine functions to model real-world occurrences such as the depth of water at various times of the day, the movement of a pendulum, etc. Scientists use exponential functions to analyze and predict the number of bacteria present after x amount of time. Scientists also use functions when analyzing the time it takes a rocket to reach a destination. Making Mathematics Relevant to Students' Lives Teachers can make mathematics relevant to students, using a variety of strategies. Teachers may include items relevant and pertinent to students within question stems, such as including 'iPad,' 'apps,' and video game names. Teachers should pose questions that are similar to what students may have asked themselves, such as, 'If I invest this much money in an account and save for x years, after how many years will I have y dollars?' Teachers should include real-world problems to solve, and not simply include rote solving of equations. Students should know what sorts of scenarios may be modeled with rational expressions. Many researchers believe that curricula should be centered on the 'real world,' with all facets of mathematics learning spawning from that center. In other words, students often know how to convert a decimal to a percentage, but when reading The Wall Street Journal, they may not be able to interpret a percentage yield. Assessment Assessment Tool A mathematics assessment tool is used to assess a student's prior knowledge, current knowledge, skill set, procedural knowledge, conceptual understanding, depth of understanding, and ability to make abstractions and generalizations. Perhaps the most important purpose of such a tool is to help the student develop and modify instruction. A teacher may determine that students are ready to surpass the current lesson or need it to be much more scaffolded. A teacher may also use the assessment to track students' progress. For example, a portfolio might show students' initial understanding of functions and end with their work with function modeling. When a teacher needs to decide on an appropriate assessment tool, he or she needs to consider the purpose of the assessment. For example, if the purpose of an assessment is to direct the instruction, a pre-test may be a good assessment to use. If the purpose of the assessment is to determine the level of student understanding, then a whole-class discussion may be desired. If the purpose of an assessment is to assess student understanding of a unit of material, then an exam would be appropriate. If a teacher wishes to analyze student understanding and ability to abstract knowledge, then a performance assessment may be used. If a teacher wishes to check off skills mastered by students, then a checklist would be appropriate. Valid Test A test is valid if it tests what it is supposed to test. In other words, a test is valid if it appropriately covers the material it is supposed to cover. For example, a topic not taught in class should not be included on a valid test. In order to construct a valid test, a teacher should make a list of all standards covered during that time period. The teacher should also closely mirror the design of problems examined in class, for homework, and in group discussions. Finally, the teacher should make sure that there is an even balance of questions to cover all of the material. Valid Exam In order to select a valid exam, a teacher should make sure that the test aligns with the objectives and standards covered in the unit. The teacher should also make sure that the test problems are similar to those covered during class time. The teacher should make sure the percentages of questions devoted to each objective are balanced. In order for a test to be valid, it must be reliable, meaning that it produces similar results with different groups. A teacher may wish to check the validity and reliability results of an exam. In general, an exam is considered invalid if it does not measure what it is supposed to measure. The exam may include questions from another unit. It may include questions with different wording techniques, making it much more difficult. The exam may include representations different from those covered in class. An invalid exam would not be reliable, meaning the results would not be consistent with different administrations of the exam. Biased questions and wording may also make an exam invalid. Assessing Students' Understanding of What Has Been Taught In order to assess thought processes, open-ended questions are needed. The teacher may wish to have students write an essay, write entries in a mathematics journal, undergo a performance task, or participate in a debate or discussion. The teacher may also design a pre-test that includes all constructed response questions. In particular, a performance task requires students to justify solutions, which provide the teacher with insight into students' understanding and reasoning. In general, the assessment should include questions that ask students to make abstractions and justify their thinking. Testing Issue Example: A student claims that an exam is more difficult and includes more content than what was presented in class. How might a teacher determine if the student's claim is true? The teacher would need to make a list of all objectives and standards covered during the time period. The teacher would also need to compile all problems and examples covered in class and as homework. Finally, the teacher would need to do a careful analysis of the wording of the problems covered in class and as homework. If any of these items are not aligned to the exam, the teacher would need to go back and re-teach the material, using the created test as a guide for instruction. Performance Task A performance task allows the teacher to assess process as well as product, meaning that a teacher can assess students' thought processes as well as their final answer. The level of student learning will be much clearer when reviewing a performance task. A performance task goes beyond a multiple-choice format, allowing for oral and tactile-kinesthetic performances. Furthermore, a performance task may combine several mathematics concepts into one assessment instrument. This type of assessment often includes real-world problems, which helps the student connect mathematics to the outside world. Formative and Summative Assessments Formative assessments are those given during the learning process. Formative assessments provide the teacher with information related to a student's progress at various stages throughout a time period. Formative assessments are used to modify instruction as needed. In other words, formative assessments inform instruction. Summative assessments are those given at the end of a learning period.
Summative assessments serve to measure the cumulative knowledge gained. Examples of formative assessments include quizzes, checklists, observations, and discussion.
Examples of summative assessments include exams, portfolios, performance tasks, and standardized tests. Four formative assessments include quizzes, checklists, observations, and discussion. Quizzes are often short assessments that may include multiple-choice items, short response items, or essay items. Quizzes are often administered following presentation of a portion of a mathematics unit. Checklists include a list of skills or concepts that should be mastered or understood. A teacher will check off all items mastered by a student. Observations are informal means of assessing students' understanding of a topic. A teacher may observe students' questions, engagement, and performance on projects. Discussion is another informal formative assessment. Discussions, both in groups and whole-class formats, allow the teacher to analyze students' thinking. Four summative assessments include exams, portfolios, performance tasks, and standardized tests. Exams may include closed-ended or open-ended questions. Exams may be administered after each unit, semester, or at the end of the year. Portfolios include tasks created by a student and may include writing pieces and other large projects. Although the portfolio contains formative work, the tool itself may be used as a summative assessment piece. Performance tasks are large-scale problems that include many different components that relate to some big idea. For example, a student may be asked to formulate a plan for modeling a real-world phenomenon with a sine function. The student may be asked to explain how the function would change, given changes to the amplitude, period, shifts, etc. The student may then explain how these components would need to change to fit a new function. Standardized tests are tests that compare a student's performance to that of other students. They are often given at the end of the school year. Scoring Rubric A strong rubric will include unique performance criteria for each bullet. In other words, a portion of one criteria statement should not be included in a portion of another criteria statement. Each criteria statement should be clearly delineated, describing exactly what the student must be able to do. Furthermore, a strong rubric will often have scoring options, ranging from 0 to 4. When designing the rubric, it is helpful to create a model student response that will warrant each rubric score. It is also helpful to provide a space to provide feedback to students. Enhancing Student Understanding In order for an assessment to enhance student understanding, it should provide an opportunity for the student to learn something. The assessment should be a learning opportunity for the student. It should prompt the student to think deeper about a mathematics topic. In other words, the student should think, 'Okay. I understand this. I wonder how the process/solution would change if I did this.' The assessment might prompt the student to ask deeper questions in the next class session or complete research on a certain topic. In order to create such an assessment, open-ended and challenging questions should be included on the exam. The exam should not consist of simple, lower-level, one-answer questions. Testing Mathematical Misconceptions In order to design such an assessment, the teacher should include mathematical error-type problems, whereby the student must look at a solution process or conjecture and determine if he or she agrees, of if and where an error occurred. The student would need to identify the error, correct it, and explain why it was erroneous. The assessment should include a variety of mathematical misconceptions. One solution process may include more than one error. A teacher may also simply ask students to participate in a collaborative learning activity, whereby the students must share ideas and thoughts regarding a new mathematical topic. Assessing Prior Knowledge Such a pre-test must not include any leading prompts. It should include open-ended and constructed-response items as well. A pre-test with solely multiple-choice items will not be sufficient, since a student has the option of guessing. The test should include higher-level questions that require connections within the field of mathematics. In other words, the questions should not all be mutually exclusive. They should build on one another. Finally, the test might include student error problems as well. Assessing Both Procedural Knowledge and Conceptual Understanding The assessment should include rote, algorithmic-type problems, as well as those that ask the student to utilize higher-level thinking, abstractions, and generalizations. The test should include open-ended, constructed-response-type problems. A performance task is an excellent assessment for assessing a student's ability to solve a problem, while also examining the student's thought processes, rationales, etc. In order to assess both types of understanding, the assessment will need to ask students to justify and explain solutions. In other words, the assessments should include questions at both ends of Bloom's Taxonomy. Pre-Test and Post-Test A post-test should be exactly the same as an administered pre-test. If the teacher is to compare the results of a post-test to a pre-test, then the test and testing conditions should be identical. The pre-test assesses students' prior knowledge, while a post-test assesses students post knowledge. Comparing the results, side by side, allows the teacher to track student progress. The teacher may wish to add additional questions to the post-test, but the original questions should remain. Assessment That Will Show What Students Do and Do Not Know The teacher should include questions that are straightforward, involve errors, require justification, and require shown work. A student self-assessment is one such tool that would show misconceptions, understood material, and advanced knowledge. The assessment should include more than multiple-choice questions. Designing a performance assessment with scaffolded questions, whereby only one solution may be found based on a previous answer, will also show students' exact level of understanding. A debate format is one type of assessment whereby the teacher will be able to see a student's level of understanding, as he or she seeks to respond with a rebuttal. Assessment to Hone in on Any Error Patterns Evident in Students' Work A portfolio would be an excellent assessment for monitoring any student error patterns. The teacher would be able to track student errors as the course progressed. The teacher would be given insight into how, and if, errors improved, or if some knowledge was acquired but other knowledge was still incorrect. The portfolio might include a series of similar questions related to a certain topic. For example, a portfolio may include function transformation questions. A student's ability to transform functions may be tracked, starting with simple linear functions and ending with complex sine functions. Components That Must Be Present in an Assessment That Supports Student Learning The assessment must require students to think deeper than what they have covered in class. It should prompt them to make connections between topics. It should invite different ways of thinking about problem solving. In other words, the student may think, 'Okay. I have seen a similar version in class. This problem is slightly different, in that the parabola is shifted left. This is the opposite of shifting right, so I will add the constant to the x-term.' The assessment will thus solidify the student's understanding of how to shift any function. Using Assessment Results of Assessments Given to ELL Learners in Order to Modify Instruction The teacher would be able to see if language itself is a barrier in learning. In other words, if the group of ELL students, as a whole, show difficulty with a mathematics topic, the teacher may deduce that the content was not clear due to minimal supporting pictures, diagrams, and auditory support. The teacher may decide to reteach the lesson, using more visual cues, verbal pronunciations, explicit vocabulary usage, and peer-group placement. Collaborative learning may be employed. Questions That a Teacher May Ask After Reviewing the Results of an Administered Exam
How Focus on Career and College Readiness Affects Assessment and Instructional Design The focus on college and career-readiness standards prompts publishers and teachers to utilize more real-world problems in instruction and assessments. The focus in mathematics classrooms is shifting to more real-world, cumulative problems that require understanding of many different mathematics concepts in order to solve. Problems are related to science, finance, medicine, etc. The focus includes the ability to apply the algorithms to many different career situations. In summary, the recent focus shifts the instructional design to an application-based status. Role of Assessment in a Classroom Focused on Cognitive Instruction A cognitively guided classroom would be similar to a constructivist classroom, in that active participation would be present. However, in a cognitive classroom (as advocated by current cognitive theorists), students are not required to invent their own knowledge. Instead, they must simply make sense of what they are observing and experiencing. They may be assisted by the teacher. Thus, the role of an assessment in such a classroom is to ascertain student thought processes. Such an assessment would ask students to describe thinking and perhaps make connections to other mathematics topics. The assessment must ascertain students' reasoning abilities. Instructional Cycle Described by a Learning Theorist The 5E Learning Model is based on the thinking of Jean Piaget. It is a constructivist learning model. Piaget believed that students construct their own knowledge via active participation and experiences. Problem solving is integral to student learning. The cycle is listed as engagement, exploration, explanation, elaboration, and evaluation. Thus, with active engagement and exploration, the student is able to develop his or her own explanation, use assimilation and accommodation to make sense of the information, and then evaluate the material and make conjectures, etc.
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