TEACHING AND LEARNING MATHEMATICS IN THE SECONDARY SCHOOL

Daniel BochicchioNeag School of Education

Assessment should

Enhance students’ learning. Be a valuable tool for making

instructional decisions. Furnish useful information to students.

Guiding Principles

Assess students before teaching, to determine their current knowledge and skill levels.

Use multiple means and measures to assess students’ performance.

Define the objectives and criteria clearly before you begin to teach the lesson or unit.

Teach students to assess their own progress through goal-setting and reflection.

Provide useful, specific, meaningful feedback throughout the learning cycle.

Assess Before Teaching

Give them a “pretest”. Observe them working on an assignment. Look at informal writing. Prepare a class discussion with the

primary purpose of determining what students know about a certain topic.

Multiple Means and Measures

Diagnostic Done before you begin actively teaching. Helps create a learning profile of each

student. Should not be graded Determines what content should be taught

first, what methods are most appropriate, and how you should organize the instructional sequence.

Formative Done during the instructional sequence in an ongoing

measure of student progress toward specific learning goals.

Allows you to adjust your instruction. Provides useful information to the students for improving

their performance. May or may not be graded. Formal: quiz, test Informal: observations, discussions, journals, questioning

Summative Done after the instructional unit is over. Evaluates the extent to which the student learned the

material you taught. Culminates in some performance or product that

measures understanding and mastery of the content standards.

Includes major tests, projects, performances Are graded.

Bloom’s New Taxonomy

Original(1956)

New(1990’s)

Remembering: can the student recall or remember the information? define, duplicate, list, memorize, recall,

repeat, reproduce state Understanding: can the student explain

ideas or concepts? classify, describe, discuss, explain, identify,

locate, recognize, report, select, translate, paraphrase

Applying: can the student use the information in a new way? choose, demonstrate, dramatize, employ,

illustrate, interpret, operate, schedule, sketch, solve, use, write.

Analyzing: can the student distinguish between the different parts? appraise, compare, contrast, criticize,

differentiate, discriminate, distinguish, examine, experiment, question, test.

Evaluating: can the student justify a stand or decision? appraise, argue, defend, judge, select,

support, value, evaluate Creating: can the student create new

product or point of view? assemble, construct, create, design, develop,

formulate, write.

Remembering (Remembering previously learned material) memorizing a formula

Understanding (Grasping the meaning of material) knowing which variables correspond to each term in the formula

Applying (Using information in concrete situations) Word problem (obtaining variables from diagrams or other data)

Analyzing (Breaking down material into parts) knowing how parts of the formula relate to each other (eg y=mx+b,

knowing how changing one term will affect the graph)

Evaluating (Judging the value of a product for a given purpose, using definite criteria) When you have finished solving a problem (or when a peer has done

so) determine the degree to which that problem was solved as efficiently as possible.

Creating (Putting parts together into a whole) Apply and integrate several different strategies to solve a

mathematical problem.

Various Types of Assessment

Portfolios Journals Notebooks Open-response questions Presentations Projects Group Projects Student Assessment Tests/Quizzes

Portfolios

A purposeful selection of work Show growth and development over time Show accomplishment Should correspond to different kinds of

assignments

Portfolio ProjectAt the end of the Third Marking Period, you will assemble a portfolio to

represent your progress in math for the year so far. A portfolio is one way of assessing what you are learning. It is a showcase of your knowledge, abilities, attitude, and understanding.

To be included: Table of Contents The test that you think shows your best work for the year. Explain why

you chose this specific test. Something mathematical that you have done this year that you are

proud of. Explain why you have chosen this item. Two problems (one from homework and one from a test) from the 3rd

quarter that you did not complete or did not answer correctly. Include a correct solution and explanation. Include the original paper.

An example of group work. Write about what the task was that you had to work on. Also, write about your role in the group and the roles of the other group members. Write about how this group work may or may not have helped you learn.

Work from two other subject areas that relate to math. Tell me how math is important to the other subjects.

One typed page reflection on personal growth in math. Let me know how your attitude towards math is, has been in the past, has possibly changed. Let me know what you think of the class. BE HONEST!! It is the best policy.

Journals

A place where students can record and reflect on their experiences.

Can express frustrations and anxieties, as well as pride

Allows quiet students to express their thoughts

Its not best to grade the entries, but read and respond to them

Sample Journal Prompts

Automathography The most important mathematical concept I

learned this week A question I still have about what we

investigated this week Group behavior or social skill I’d like to see

improve in class What did you learn about”____” today Explain the following concept in your own words.

Performance Assessments

Let the students measure or construct Observe them perform a task Can be assigned individually or as a

group

Designing and Flooring a HomeThe Design Using graph paper, you will design the blueprints to a home. Each square unit will represent one square foot.

Be realistic with your measurements. You would not have a master bedroom that is 5’X 30’. Leave enough room for a door (3 feet is a standard single door).

Think about the layout of the house. The dining room and kitchen should NOT be on opposite sides of the house. The only way into the Family room should NOT be through the master bedroom. You might need hallways and walls added to separate rooms.

Label and Scale Please make sure to label each room (Kitchen, Living Room, Game Room). If you have multiple rooms that are the same type, label them as Bathroom 1, Bathroom 2, etc. Label the dimensions of each side of each room. Be sure to include the correct unit (feet). Calculate the area of each room and circle this number. Be sure to include the correct unit (sq. ft).

Be Clean and Neat Straight lines need to be straight. Use a ruler. No Sloppiness

Choose the Type of Flooring and Calculate the Cost. For each room, you will need to choose the type of flooring. You should be realistic about the type of flooring. Rug in the bathroom?? Do not repeat the same exact flooring between rooms. You can have rug in the living room and game

room, but choose different types. Fill in the table on the back of this page.

Open-Response Questions

Can have more than one correct answer May have a BEST solution, but different

paths of reasoning can be followed to arrive at the correct solutions.

Very CAPT-like.

Farmer Ben has only ducks and cows. He cant remember how many of each he has. He does know he has 22 animals, which is his age. He also remembers that those animals have a total of 56 legs, which is his father’s age. Assuming each animal is normal, how many of each does Farmer Ben have?

Presentations

Individual students or groups can present Teacher and student involvement in

evaluation. Discuss in advance the keys to a good

presentation. Get student input. Can present to a small group or the whole

class. Prepares for their Grad project

Math History PresentationYou will be asked to work on a presentation

concerning a famous mathematician. This presentation will want to address the following points: The name of your mathematician Describe their general life. (i.e. childhood, schooling,

family, etc.) Personal strengths/weaknesses; character flaws Describe the time period in which they lived Discuss their mathematical work/accomplishments (to

the best of your ability) Requirements

Should last 3-5 minutes (no more…no less) Have a visual (i.e. posterboard, video, handouts) Your

choice.

List of Mathematicians

Euclid of AlexandriaHypatia of Alexandria Apollonious of Perga

Thales of Miletus Blaise Pascal Leonhard Euler Isaac Newton Johann Bernoulli Nikolai Lobachevsky Grace Hopper Alan Turing Andrew Wiles Gottgried Leibniz Rene Descartes Felix

Klein G. F. Riemann George Dantzig Albert Einstein Emmy Noether Leonardo Fibonacci Pythagoras of

Samos Karl Gauss Sophie Germain David

Hilbert Arthur Cayley Archimedes of Syracus Kurt Godel John Napier Pierre Fermat Omar Khayyam

Projects

A multi-step assignment students can complete both inside and outside of the classroom

Involve a series of related investigations, problem-solving situations, research demonstrations.

Increase student understanding and enthusiasm

Exploring the United States  You are a reporter for a new travel book, Exploring the United States.

You have been assigned by your editor to write in this guidebook. You will need to describe the city and let the readers know what activities and attractions that city has and during what time of year it would be best to visit them.

  You will also be including an analysis of the city’s temperature cycle for

the year. Your editor knows of your extraordinary trig ability and wants you to illustrate your article with a graph of the temperature cycle and to give the equation of the temperature cycle.

  You will need to create a brochure describing interesting facts about

your city. You should also include information regarding different activities and attractions the city has to offer. You should use Excel to create the graph.

One will need to hand in one brochure AND one graph (with its equation).

"Square Dancing" Sally invited 17 guests to a dance party. She assigned each guest a number from 2 to 18, keeping 1 for herself. At one point in the evening, Sally noticed the sum of each couple's numbers was a perfect square. Everyone was wearing their numbers on their clothing. The question is how is each couple numbered and what was the number of Sally's partner?  

Please type 1-2 paragraphs explaining your process for getting your final answer. The beginning of the paper should restate the question in your own words, and should then be followed by your explanation and final answer. Please use correct spelling and grammar. Clearly mark your name and teacher on your paper, and staple it to the assignment sheet.

Creative Trigonometric Writing  This project is designed to help you fully grasp the concept of using trigonometry to solve problems. This includes

finding the length of a missing side or a missing angle of a right triangle, identifying these problems in everyday life, drawing them, calculating their values, and weaving it all into an interesting, amusing or creative story!

Your task is to develop a story involving a missing angle AND a missing side of a right triangle AND an arc length. To support your story, you will need to carefully draw a picture illustrating the problem in your story. You will need to include all formulas and all of the steps required to come up with your answer for your story.

Requirements     a)      A Creative, Unique, and Imaginative Story i)         Imaginative ii)       Appropriate subject iii)      Must be a trigonometry problem (finding a missing side AND angle) iv)     The story MUST be more than one sentence. It must be a STORY. v)       It must be creative!  b)      A Drawing i)       The drawing must be visible and clearly definable

c)       A diagram of the right triangle and circle to solve the three problems i)         Provides a clear representation of the problem ii)       Include realistic measurements iii)      Include units iv)     Neat (MUST use straight edge) v)       It can be incorporated into the drawing or picture (above) but it must be clearly legible and visible vi)     It must be a right triangle, and labeled with a right angle mark

d)      Calculations i)         Show formulas you used ii)       Show all steps iii)      Include units in your answer

e)       An appealing presentation i)         Professional look ii)       Story should be neatly written. iv)     Appropriate subject v)       Title vi)     Size limit: At least one full page.

Group Work and Projects

Forms cooperative efforts Allows students to gain new perspectives

on learning and thinking Forces you to monitor work carefully Allows students to be held individually

accountable (self- and group-evaluation)

Operation: Popcorn...what's outside is what counts.

You are a package designer for a popcorn company. The company needs you to design a "box". You need to minimize the cost of the packaging material, as well as consider uniqueness, stackability and other non-monetary concerns. Packaging material will cost 1/4 cent per sq cm. Package to ship two cups of popcorn.

As a two-person team, research/calculate as many designs as possible. You will submit a minimum of two (2) design proposals for final approval. One design MUST be a cylinder.

Prepare a report for your supervisor detailing all design proposals. This report should be well justified and your decisions must be backed up with either calculations or words. You must write in complete sentences. The designs should be well prepared and neat. When submitting a proposal there should be no “free-handing” of lines. Work on this as a team and split the work accordingly. If issues arise, discuss them and come up with a plan. Include: Flat patterns (net) Surface areas (and calculations) costs (and calculations) pros and cons recommendations

Student Assessment

Have a student correct their own test or quiz using a rubric How do you correct?

Have a student correct a “student-generated” test or quiz using an answer key How do you analyze student work? How would you award credit? Full vs. Partial How do you correct student mistakes?

Is the following student’s work correct? If a mistake exists, then in paragraph form:

Identify where the mistake occurs. Describe the incorrect assumption presumed

by the student. Describe how the student could have avoided

or corrected the mistake.

Tests/Quizzes

Is your test in-line with your objectives? Does the quiz model what was done in

class? Does the textbook teach your course? Do students know what to expect?

Open-Ended Items

Open-ended items can be answered using a variety of strategies and may have a variety of answers. In general, open-ended items are more complex in nature than grid-in items and each open ended item should require about 5 – 8 minutes to complete. The answers to open-ended items are scored holistically, using a four-point generic rubric and anchor papers.

Rubrics are used to assess student understanding on a continuum and are a common way to score student work on open-ended tasks and complex problems. One key aspect to using a rubric is to let the students know in advance the criteria that are being used to assess their work. The rubric helps to clarify the expectations of the teacher for all levels of performance.

Score 3The student has demonstrated a full and complete understanding of all concepts and processes essential to this application. The student has addressed the task in a mathematically sound manner. The response contains evidence of the student’s competence in problem-solving and reasoning, computing and estimating, and communicating to the full extent that these processes apply to the specified task. The response may, however, contain minor arithmetic errors that do not detract from a demonstration of full understanding. Student work is shown or an explanation is included. 

Score 2The student has demonstrated a reasonable understanding of the essential mathematical concepts and processes in this application. The student’s response contains most of the attributes of an appropriate response including a mathematically sound approach and evidence of competence with applicable mathematical processes, but contains flaws that do not diminish the evidence that the student comprehends the essential mathematical ideas addressed in the task. Such flaws include errors attributed to faulty reading, writing, or drawing skills; errors attributed to insufficient, non-mathematical knowledge; and errors attributed to careless execution of mathematical processes or algorithms.

 Score 1

The student has demonstrated a partial understanding of some of the concepts and processes in this application. The student’s response contains some of the attributes of an appropriate response, but lacks convincing evidence that the student fully comprehends the essential mathematical ideas addressed by this task. Such deficits include evidence of insufficient mathematical knowledge; errors in fundamental mathematical procedures; and other omissions or irregularities that bring into question the extent of the student’s ability to solve problems of this general type.

 Score 0

The student has demonstrated merely an acquaintance with the topic. The student’s response is associated with the task in the item but contains few attributes of an appropriate response. There are significant omissions or irregularities that indicate a lack of comprehension in regard to the mathematical ideas and procedures necessary to adequately address the specified task. No evidence is present to suggest that the student has the ability to solve problems of this general type.

One way to use the rubric is to consider it as a two-step decision-making process

Is the work acceptable?

YES

YESYES

NO

NONO

Does the student show full understanding?

Does the student show some understanding?

3 012

The diagram shows a hot air balloon tied to the ground by a rope. The balloon will be used in the Thanksgiving Day parade. To the nearest meter, what is x, the distance from the balloon to the ground?

91 meters

70 meters

On Friday, the following equation gave the exchange rate between the value of the Canadian dollar (C) and the U.S. Dollar (U):

 On Saturday, the exchange rate had changed

to the following equation:

On Friday, Jeremy changed $30 U.S. to Canadian dollars based on Friday’s exchange rate. He did not spend any of the money, and on Saturday he changed it back to U.S. dollars at Saturday’s exchange rate. How much money, in U.S. dollars, does Jeremy now have?

A diagram of a racecar track is shown below. The length of the straight-away section is 400 yards. The ends are semicircles with a diameter of 100 yards. Find the total length of one lap of the track.

Carrie bought her mother a Lily bulb, a plant that is supposed to bloom when it reaches a height of 38 centimeters. She noted the height of the plant at the end of each day, as shown in the table below. If the plant continues to grow as shown in the table, on what day will it bloom?

Day Height at the End of the

Day (centimeters)

1 22 33 54 85 12

Michelle plans to rent a jet ski one day this summer. She is going to use a rental service that charges a fixed fee of $45 plus $13 for each hour or part of an hour that she uses the jet ski.

Fill in the table of values that shows the Total Rental Charges

for the Number of Hours.

Then, extend the pattern to determine the total charges if Michelle rents the jet ski at 8:45 in the morning and returns it at 6:35 that evening. Show your work and explain how you found your answer. Number of Hours Total Rental Charges

0-1

1-2

2-3

3-4

4-5

Approximate the total area of Connecticut in square miles, using the map and the given scale.

Show your work and explain how you found your answer.

  1 cm = 1.2 mile

William is planning to market a new kind of jellybean. He plans to sell the jellybeans in containers like the ones above with the dimensions as shown. Consumers can buy a small, medium, or large container of jellybeans. Which is the best deal in terms of cost per cubic centimeter? Show all work and explain how you found your answer.

During a 10-day period, the daily high temperatures in Vernon were: 

45 F, 10 F, 40 F, 51 F, 45 F, 49 F, 42 F, 53 F, 57 F, 55 F

a. Find the mean (average) and median for the set of temperatures.

 b. Which one is a better measure of central

tendency for the given temperatures? Explain your decision.

c. The temperature on the11th day was 92 F. Which measure of central tendency would change the most? Explain your choice.