tab 6: assessment table of contentstab 6: assessment: table of contents 6-i tab 6: assessment table...

Mathematics TEKS Refinement 2006 – 9-12 Tarleton State University

Tab 6: Assessment: Table of Contents 6-i

Tab 6: Assessment Table of Contents

Master Materials List 6-ii

Opening Activity – Assessment Pyramid 6-1

What’s Your Problem? 6-10 Handout 1-Sample Assessment Items-Roger Throws a Ball 6-25 Handout 2-Three Problem Types Labels 6-27 Handout 3-What’s Your Problem? Items 6-30 Handout 4-Three Problem Types – How to Write 6-58

The Power of Creating 6-59 Handout 1-Sample Class Responses 6-63 Transparency 1-Slope of 4 6-64 Transparency 2-Y Intercept of 4 6-65

Closing Activity –Assessment Should Drive Instruction 6-66

Resources 6-72


Tab 6: Assessment: Master Materials List 6-ii

Tab 6: Assessment Master Materials List

Blank paper Chart paper or white board space Current textbook(s) or local curriculum (optional) Copies of all the PowerPoints with space for note taking What’s Your Problem? Handouts Generation Power Handout and Transparencies The following materials are not in the notebook. They can be accessed on the CD through the links below.

PowerPoint: Opener

PowerPoint: What’s Your Problem? PowerPoint: The Power of Creating PowerPoint: Closer



Activity: Opening Activity – Assessment Pyramid Overview: Assessment happens in many forms of classroom interaction:

questioning, homework, quizzes, tests, projects, classroom tasks. All of these forms of assessment have their own important place in the mathematics classroom. They all have in common the idea of ascertaining what students know and what they do not know. These assessments can be thought of in three dimensions: level of reasoning required, level of difficulty, and degree of skill or conceptual understanding required. The Assessment pyramid can be helpful in providing language and a perspective when looking at specific items, at a complete test, and at an entire course. Over time, all classroom assessments should generally fill the pyramid. The pyramid is not a rectangular prism, suggesting equal amounts of low and high level questions because it takes fewer high level reasoning questions to assess mathematical understanding. It takes more low level reasoning questions to assess mathematical understanding.

Participants will examine and discuss the Assessment Pyramid. They will

consider several assessment items and their approximate positions on the pyramid. Participants will reflect on their own assessment practices and consider several guiding questions, including how to change their existing questions. This activity should prime them for the next section – how to change questions.

Materials: PowerPoint: Opener

Copies of the PowerPoint with space for note taking

Grouping: Tables of 4 Time: 30 minutes Lesson: Distribute copies of the PowerPoint to help participants focus on important

ideas in the presentation as they take notes. Show the PowerPoint presentation Opener. Use the following notes pages for elaboration.

Procedures Notes Slide

1

Assessment

Mathematics TEKSRefinement Project




2 AssessmentIntroduce the Assessment Pyramid (adapted from de Lange’s Assessment Pyramid.) Note that the pyramid is a graphic that helps identify qualities or properties or dimensions of assessment items. Point out the two primary sections:

• concepts • mathematical skills

Level of difficulty runs from front (easy) to back (difficult). Levels of reasoning go from bottom (reproduction) to top (higher level). The line separating concepts and skills gets thinner as you move up the pyramid, suggesting that the lines blur between concepts and skills as you raise the level of reasoning required to solve problems. The next few slides give the opportunity to discuss the levels of reasoning in more depth. Be very clear that participants should not get bogged down with the pyramid. It can be useful and helpful but it should not cause arguments through the rest of the training about just how high or deep an item should be placed, etc. Its purpose is to get the conversation going and provide some vocabulary and perspective.

Slide 3 Assessment

Lower level - reproduction,procedures, concepts,definitions

Throughout the discussion, be sure to juxtapose the ideas of low level to high level reasoning versus easy to difficult. Many teachers talk past each other by using the same words to discuss different qualities of assessment items. Some teachers say, “I do not understand why my students are not successful. I ask them really hard questions. I do not know how to better prepare them for the hard/difficult questions they see on TAKS or other tests.”



Procedures Notes But is there a difference between hard or difficult questions and questions that require a higher level of reasoning? Now look at each level of reasoning separately. Lower level reasoning items “deal with knowing facts, representing, recognizing equivalents, recalling mathematical objects and properties, performing routine procedures, applying standard algorithms, and developing technical skills, as well as dealing and operating with statements and expressions that contain symbols and formulas in ‘standard’ form. Test items at this level are often similar to those on standardized tests and on chapter tests related to conventional curricula. These are familiar tasks for teachers and tend to be the types of tasks they are able to create.” (Romberg, 17)

Slide 4 Assessment

Middle level - connectionsand integration for problemsolving

“At this level, students start making connections within and between the different domains of mathematics, integrate information in order to solve simple problems, have a choice of strategies, and have a choice in the mathematical tools. At this level, students also are expected to handle representations according to situations and purpose and need to be able to distinguish and relate a variety of statements (eg., definitions, claims, examples, conditioned assertion, proofs). Items at this level often are placed within a context and engage students in mathematical decision making. These tasks tend to be open and similar to instructional activities.” (Romberg, 18) Note that at this level, the lines blur between concepts and skills. It is more about connections. Also, it takes fewer middle level reasoning items to assess mathematical achievement. A student’s answers to items higher up the pyramid



Procedures Notes give a more complete picture of that student’s understanding and skill. Therefore teachers need less higher level reasoning questions to ascertain a student’s mathematical achievement. Also, those higher level reasoning items generally take more time and involve more work.

Slide 5 Assessment

Higher level -mathematization,mathematical thinking,generalization, insight

“At this level, students are asked to mathematize situations: recognize and extract the mathematics embedded in the situation and use mathematics to solve the problem; analyze; interpret; develop models and strategies; and make mathematical arguments, proofs, and generalizations. Items at this level involve extended-response questions with multiple answers.” (Romberg, 18) Over time, a complete assessment program should “fill” the pyramid.

Slide 6 Consider the following:

A rectangular prism is 2cm x 4cm by 6cm. One dimension isenlarged by a scale factor of 3. What is the volume of the enlargedfigure?A rectangular prism is 2.7cm x 0.45cm by 609.01cm. Onedimension is enlarged by a scale factor of 3.5. What is the volumeof the enlarged figure?When a figure is dilated by a scale factor k to form a similar figure,the ratio of the areas of the two figures is ___ : ___ .A certain rectangular prism can be painted with n liters of paint.The factory enlarged it by a scale factor of 3 to make a similarprism. How much paint do they need to paint the larger box?

Ask participants to consider the four items on Slide 6. How would they rate the items? Easy, hard? What other kinds of words would they use to describe the items? Have them discuss in their groups.

Slide 7 Assessment Items - Where?

Where might the items fit in the pyramid?




8 Assessment Items - Where?A rectangular prism is 2cm x 4cm by 6cm.One dimension is enlarged by a scale factorof 3. What is the volume of the enlargedfigure?A rectangular prism is 2.7cm x 0.45cm by609.01cm. One dimension is enlarged by ascale factor of 3.5. What is the volume of theenlarged figure?When a figure is dilated by a scale factor k toform a similar figure, the ratio of the areas ofthe two figures is ___ : ___ .A certain rectangular prism can be paintedwith n liters of paint. The factory enlarged itby a scale factor of 3 to make a similar prism.How much paint do they need to paint thelarger box?

Ask participants to consider where each of the four items might sit in the pyramid. Have them discuss this with their group. Then ask groups to share their thinking with the whole group. #1 - Find the area is a fairly easy skill question with lower level thinking needed. #2 – This question is still a skill question with lower-level thinking but it is difficult because of the crazy numbers. Do we as teachers sometimes confuse “difficult” with higher-level thinking? Do we sometimes create difficult question because of the computation and fail to create higher level thinking questions? #3 – This could be considered a concept question but one that only requires memory and therefore easy and low level. This does depend highly on how the concept has been taught. If students have not had explicit experience with abstracting this concept – thus they would have to do that here in this item, that would change the item to being a much higher level problem. #4 – This requires a higher level of thinking and a conceptual knowledge of the fact that painting requires the knowledge of surface area. If a student has a good understanding of scale factors and the result to area, this is actually a fairly easy question.

Slide 9 Your Assessment Items - Where?

Teacher questioning?Homework?Quizzes?Tests?

Assessment is a broad term that for many has different implications. Is it all about grades? Is it a continual process that informs instructional decisions? At this point, ask participants to consider what they deem “assessment” and where their assessments might fit in the pyramid. While assessment includes teacher



Procedures Notes questioning, homework, quizzes, tests, and more, the discussion that follows will focus on individual assessment items - specific questions, tasks, problems - and how teachers can differentiate where the items are on the pyramid so that teachers can make better assessment decisions.

Slide 10 Guiding Questions

How can I ask questions for which students can not justmemorize their way through? How can I ask questionsthat demand that students actually understand what isgoing on?How can I ask questions that students can learn fromwhile answering?How can I make sure that I have higher level reasoningquestions and not just computationally more difficultquestions?

Ask participants to consider these guiding questions as we continue.

Slide 11 Passive Assessment Expertise

Understanding the role of the problem contextJudging whether the task format fits the goal of theassessmentJudging the appropriate level of formality (ie., informal,preformal, or formal)Judging the level of mathematical thinking involved inthe solution of an assessment problem

Feijs, de Lange, Standards-Based Mathematics Assessment in Middle School

The goal of this assessment discussion is to help teachers to select the assessment items that fit their needs and purposes, as they consider their entire assessment program. It is not to teach teachers to create such items, but to judge the appropriateness of those items from which they are selecting.

Slide 12 The Assessment Principle

Assessment should become a routine partof the ongoing classroom activity ratherthan an interruption.

NCTM’s Principles and Standards for School Mathematics (2000)

Before participants try to answer the guiding questions together, look at the Assessment Principle from the NCTM standards. Assessment should be a routine part, not an interruption. How can participants do that better? The next activities will give participants some ideas.



Procedures Notes Slide 13 TAKS Item 9th grade 2004

Tony and Edwin each built a rectangulargarden. Tony’s garden is twice as longand twice as wide as Edwin’sgarden. If the area of Edwin’s garden is600 square feet, what is the area ofTony’s garden?

Here is a TAKS released item from 2004 – Ninth grade. Ask participants: Which of the items that we looked at would better prepare students for this item? Specifically, would the computationally more difficult question better prepare students for this more abstract TAKS question? Or would the more open ended, higher level thinking question better prepare students for this question? It is not possible to predict all of the ways that a TEKS will be assessed. Would higher level thinking questions open the door for students to at least be thinking in that direction?

Slide 14 Our focus

Think about current classroomassessmentsHow can they improve?

Again, focusing on classroom assessment items we raise the question, “How can they improve?” Don’t answer this yet.

Slide 15 Take one typical assessment

What is the purpose of the assessment?Where are the items in the pyramid?Are you satisfied with the balance?

Here Trainers could have participants choose one of their classroom’s assessments (assignments, quizzes, tests) and answer these questions. This could be a group activity where the group looks at a common assessment, perhaps a textbook assessment or shared exam. Trainers could also have each teacher individually select a quiz he/she has given and then share with others an example of an item from 3 different places in the pyramid.



Procedures Notes Slide 16 Changing existing questions

to higher leveling reasoningto concept questionsmaintain balance between concept andskill questionsshift focus from what students do notknow to what they do know

This is a transition slide to get participants primed for the next activity, which is changing and improving items.

Slide 17 Targeted Content

(A.2) Foundations for functions. The student uses theproperties and attributes of functions. The student isexpected to: (D) collect and organize data, make andinterpret scatterplots (including recognizing positive,negative, or no correlation for data approximatinglinear situations) , and model predict, and makedecisions and critical judgments in problem situations .[vocabulary of zeros of functions, intercepts, roots]

Note that the items and discussion will be based on targeted content - the TEKS listed in the following slides.


(A.4)(C) connect equation notation withfunction notation, such as y = x + 1 andf(x) = x + 1.


(G.11) Similarity and the geometry ofshape. The student applies the conceptsof similarity and justifies properties offigures and solves problems. The studentis expected to: (D) describe the effect onperimeter, area, and volume when one ormore dimensions of a figure arechanged and apply this idea in solvingproblems.



Procedures Notes Slide 20 Targeted Content

(G.6) Dimensionality and the geometry of location. Thestudent analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses theserepresentations to solve problems. The student isexpected to: C) use orthographic and isometricviews of three-dimensional geometric figures torepresent and construct three-dimensional geometricfigures and solve problems.

Slide 21

So, let’s look at some ways toimprove …

Slide 22 Consider the following:

Factor: x2 - 5x - 6Factor: 36x2 + 45x - 25A soccer goalie kicks the ball from the ground.It lands after 2 seconds, reaching a maximumheight of 4.9 meters. Write the function thatmodels the relationship (time, height).Define “root” of an equation.

Here is the optional beginning for algebra teachers.

Resources: Romberg, Thomas A. ed., Standards-Based Assessment in Middle

School: Rethinking Classroom Practice. New York: Teachers College Press. 2004.

Verhage, H., & de Lange, J. (1997, April). Mathematics education and

assessment. Pythagoras, 42, 14-20.


What’s Your Problem? 6-10

Activity: What’s Your Problem? Overview: Examination, discussion, and writing of three types of assessment items:

snapshot problems, un-doing problems, and error analysis problems. Materials: Handout 1-Sample Assessment Items-Roger Throws a Ball

(pages 6-25 – 6-26) Handout 2-Three Problem Types Labels, 1 for large group

(pages 6-27 – 6-29) Handout 3-What’s Your Problem? Items (pages 6-30 – 6-57) Handout 4-Three Problem Types –How to Write (page 6-58)

PowerPoint: What’s Your Problem? Copies of the PowerPoint with space for note taking

Grouping: Groups of 3 Time: 1.5 hours Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation What’s Your Problem? Use the following notes pages to elaborate on the content of each slide.


1

What’s Your Problem?

The purpose of this PowerPoint is to give participants examples and experience with alternate problem types.

Slide 2 Ways to Modify Questions

Given limited timeFocus on three categoriesNot the only onesPrompt other methods

Discuss the disclaimers: We only have a limited amount of time. Therefore we are going to focus on three categories. They are certainly not the only ways to turn lower level thinking, closed questions into higher level thinking questions – there are other ways for sure, but these are helpful



generalities to look at. They are a place to start. This discussion will certainly prompt other methods and that is also one of the goals – to get teachers to consider alternatives.

Slide 3 Three Ways to Modify Questions

Un-DoingError AnalysisSnap Shot

There are three possibilities we will refer to loosely as “Un-doing,” “Error Analysis,” and “Snap Shot.” These are not technical names, just general descriptors of broad categories, again with the intent of giving teachers alternatives.

Slide 4

Examples of the Three Types

Slide 5 A Typical Textbook Item

Solve for x:

x2 + 9x − 36 = 0

A typical item is a quadratic equation, and students are asked to solve for x. Do students need to be able to do this? Of course. What other kinds of questions could we ask so that students learn more about quadratic equations and solving them?



Slide 6 What might it look like ….

as an Un-Doing problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 7 Un-Doing Example

Write and graph 2 quadratic functionsthat have zeros x = -12, x = 3

Have participants briefly discuss this problem with a partner or group. In mathematics, we often do something and then un-do it. We multiply binomials, we factor the product. We add, we subtract. Here, instead of solving a quadratic equation to find the zeros of the function, we give students the zeros and ask them to go backwards to find and graph the functions. If they can do it one way, often it is a good assessment to see if they can back up, un-do the process, start with the answer and work back to find the problem.


Write and graph a quadratic function thathas

a. one x-intercept.b. two x-intercepts.c. no x-intercepts

Here is another “Un-Doing” example. Have participants briefly discuss it. Ask them to consider how a student might think differently to solve these problems. What extra or additional parts about quadratics might be embedded in this problem beyond what students have to think about to do the original “solve for x” problem? Did this and the previous example seem more open? Might this possibly allow the teacher to see a greater variety of solutions and strategies that when discussed can build strength in connecting the different approaches?




as a snap shot problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 10 Snap Shot Example

Yesterday in class we solved some equations graphically. What was the equation we were solving below? What was the solution?

The TEKS call for multiple representations and for the use of graphing technology. How does one assess this? One example is to take a snap shot of your classroom instruction and ask about it. The day before this example was presented, the class had used graphing technology to solve quadratic equations like x2+9x-36=0 by graphing y1= x2+9x-36 and y2=0 and finding the intersection. Were students just pushing buttons, memorizing a series of steps, or were they looking at the graph and realizing that they were looking for the x values for which the y’s were 0? One way to assess would have been to give them more equations to solve. Another example is to give them the graphing windows and ask about them. This shifts the focus from button pushing to connecting.


Chelsea dropped her homework in a puddle.Help her reconstruct the blurry areas. Whatquestion was she answering?

Another way to open up the discussion is to take a procedure and hide carefully selected parts of it. This way, students have to think about someone else’s strategies. This is a way to see if students really know what is going on - the reasons for the steps - or if they are stuck in one way of “doing” it. This snap shot example is also a way to assess a student’s understanding of a



manipulative model without the teacher having to stand over the student’s shoulder and watch him/her use algebra tiles.


as an error analysis problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 13 Error Analysis Example

Ralph:Factor

Eleanor:Solve:

Sanna: Solve

x2 + 9x − 36 = 0(x +12)(x − 3) = 0x = −12, 3

x2 + 9x − 36x2 + 9x − 36(x +12)(x − 3)

x2 + 9x − 36 = 0

x2 + 9x − 36 = 0x2 = −9x + 36

x2 = −9x + 36x = −3, 6

Each of these three examples has a different commonly made error. Students are asked to examine these erroneous processes and find the error(s). Ask teachers to consider the common errors of their own students. Suggest that instead of only re-teaching the correct method, they might also consider asking students to analyze the common errors made in their own classrooms. Here we see that Ralph solved instead of factored, Eleanor factored instead of solved, and Sanna’s first error was to think that the square root of a sum was the sum of the square roots.


Diane: Graph

x2 + 9x − 36 = 0(x +12)(x − 3) = 0x = 12,−3

y = x2 + 9x − 36

Another commonly made error: Diane did not set each factor equal to zero to find the value of x when the product equals zero.



Slide 15 TAKS Item (9th grade 2004)

The area of a rectangle is 3x 2 + 14x + 8,and the width is x + 4. Which expressionbest describes the rectangle’s length?

3x + 22x + 42x + 23x − 2

a.

b.

c.

d.

Here is a released item from the 2004 9th grade TAKS. Ask participants to consider the various problems at which they have just looked and how these problems might have prepared students for this item and the next two. Compare the original, typical question’s place on the pyramid with the locations of the other questions.


What are the roots of the functiongraphed below?

a. (-1, -9) and (0, -8)

b. (0, -4) and (2, 0)

c. (-4, 0) and (2, 0)

d. (0, 2) and (0, -4)

For extra credit, ask participants to find the error in the wording of this TAKS question.


Which ordered pair represents one of the rootsof the function ?

a. (-5/2, 0)

b. (-4, 0)

c. (-5, 0)

d. (-20, 0)

f x( )= 2x2 + 3x − 20

For extra credit, ask participants to find the error in the wording of this TAKS question.

Slide 18 Earlier with Roger

Earlier we looked at a series of graphs andtables that modeled Roger throwing abaseball upward from a downward movingelevator. Based on that work, answer thefollowing. The graphs and tables belowrepresent Roger starting from a differentheight, throwing at a different initial velocity.

A more effective discussion of assessment begins with a common experience to discuss. After participants have done the Roger Throws a Ball activity (page 4-15), they can now have a rich discussion about how to assess it. Ask participants to brainstorm how they might assess the Roger Throws a Ball activity.




1. Match the graph with the appropriate table.

2. What are the roots, solutions shown?

Here are several possibilities, ranging from low level to mid level reasoning questions. Ask participants to generally categorize the problem as Un-Do, Error Analysis, Snap Shot or other. (These examples are provided in Handout 1-Sample Assessment Items-Roger Throws a Ball, (pages 6-25 – 6-26) 1 and 2: These can be considered Snap Shots with an un-doing feel: take a snap shot of the activity, and ask students to un-do, or reverse, the procedures they did in Roger Throws a Ball.


3. When will Roger and the baseball be at the same height?

4. Equations have ____ or ____, whereas functions have

______ or _______.

3: Snap Shot. Were students paying attention in class, and did they make the necessary connections during the activity? 4: Low level vocabulary recall. Some things just have to be memorized. Ideally, teachers give students enough experience with concepts that the labels become automatic, but they are labels none the less and must be memorized.


5. What is Roger’s new starting height?

5: Snap Shot. Were students paying attention in class, and did they make the necessary connections during the activity?


Yesterday we looked at a series of graphs and tables thatmodeled Roger throwing a baseball upward from adownward moving elevator. Based on that work, Rauland Marco answered the following question. Who isright and why?

1. Given the table and graph representing Roger throwinga baseball upward from a downward moving elevator, whatare the roots, solutions shown?

RaulRoots and solutions are the same as zeros.Y1 is zero at x = 0 and in between x = 3and x = 4, so the roots are x = 0 and x isabout 3.5

MarcoA roo t is the so lution t o an equa t ion sow e are l ooking fo r w here Y1 = Y2 . I nt h e t ab le , both Y1 and Y2 have t he sameva lue s a t x = 0 and x = 4 . On the g r aph ,Y1 and Y2 in t er sec t at the se sa me xva lue s x = 0 and x = 4 .

Error Analysis




Bo was absent yesterday. When you startedtelling him about the Roger-throwing-the-ball-elevator activity, he said, ŅWhatÕs the big deal?Roots, zeros, solutions, x-intercepts Š theyare all the same thing.ÓHow do you respond to Bo?A complete answer includes graphs, tables,equations and discussion.

Error Analysis: This is a fine example of a problem where one can see more clearly what a student knows, instead of just what the student does not know. Wrap up: Teachers need all kinds of assessment all over the pyramid. These items have just provided a look at three broad categories that might help participants open up their options. These three categories should not be considered the only kinds of assessments. Trainers should point out that looking at these three categories should only just be considered a way to get the conversation started.

Slide 24 Part 2: As A Class

Everyone should have one problem fromthe setDiscuss the problems in your group.Decide where the items would best fit.Post your problemGallery walk - do you agree?Choose one to discuss as a group

Have participants look at some more items to continue to get a better sense of ways to alter and adjust classroom assessment for better student learning. Use Handout 2 (pages 6-27 – 6-29) to label sections of the room as Un-Do, Snap Shot, and Error Analysis. Distribute the problem items (Handout 3 pages 6-30 – 6-57), one per person if possible. Have participants work as a group and post each item under previously labeled sections of the room (Un-Do, Snap Shot, and Error Analysis). After items are posted, participants consider if they agree as they take a gallery walk. For items they think are posted incorrectly, participants could flag them with a red flag. The purpose here is not so much the three problem types, it is more to expose participants to alternative ways to assess. As a whole group, discuss the groupings. Bring out the following points: Un-Doing: Much of mathematics is doing something and then un-doing it. Many times a great question to assess if students



“got it” when doing something is to ask students to un-do it, to back up from the answer, to come at it from a different direction or representation. Some of the Un-Doing questions are a specific type – a creating type. This is a fine time to discuss this type that, for our purposes, is included in the Un-Doing group. Note: In the creating type of Un-Doing questions, students are asked to create or generate different answers. Posing questions where the answer becomes the question opens up the social “space” in the classroom to allow all students the opportunity to participate and makes them accountable for the content they are learning. “Generative design centers on taking tasks that typically converge to one outcome and turning them into tasks where students can create a space of responses.” Stroup, Ares, Hurford, 2005 Error analysis: Taking common misconceptions and mistakes and putting them up front for students to consider and explain. Snapshots: Taking a snapshot out of the middle of a process or solution or activity and asking students about it.

Slide 25 Discussion

Un-DoingError AnalysisSnap Shot

Have groups share out the one or two problems they found most interesting or compelling. Use the red flags, if any, to generate more conversation about the problems. It is less important if everyone agrees on the type of item. It is more important to discuss how the items assess student thinking.



Slide 26 Advantages and Disadvantages

GradingConceptual understandingMemorization

If they have not come up already, briefly discuss the advantages and disadvantages of these kinds of assessment items (Un-Do, Snap Shot, and Error Analysis). One disadvantage:

• Grading - many of the problems are more open ended. This may be a barrier for participants who have little experience or few resources to deal with more open-ended questions. In the Closer section there are resources to help.

Two advantages: • Conceptual understanding: These

assessments demand more conceptual understanding than many typical textbook bare problems.

• Memorization: Students cannot just memorize their way through these problems. They actually have to know what is going on.

Slide 27 Write your own

Choose a TEKS statementWrite a typical question to assess it.Write it as an Un-Doing questionWrite it as an Error Analysis questionWrite it as a Snap Shot question

Distribute Handout 4-Three Problem Types: How to Write (page 6-58) and blank paper. With participants in groups, have them create some assessment items on the blank paper using the How to Write handout. Participants can consult their textbooks for typical questions. They can also do a “search and rescue” as they search the textbook for examples of the three types of items. Have each group choose one or two and share out with the whole group. Collect, copy, and then hand them out so that everyone in the group will have more examples. Ask participants to label the items in some way as to suggest where they might use each item (on which assignment, test, project, what time of the year, etc.)



Slide 28 Snap Shot Problems

What are two ideas, processes, orrepresentations that students mix up?Juxtapose them and ask which is which.What part of a large activity can you grabto assess if students got the gist of thelarge activity?

Trainers might show this slide while participants are sharing Snap Shot problems they created.

Slide 29 Un-Doing Problems

Can you start with the answer?Can you start in the middle?Can you change one constraint?Can you start with a differentrepresentation?Ask students to create or invent thebeginning of a problem.

Trainers might show this slide while participants are sharing Un-Doing problems they created.

Slide 30 Error Analysis

What are the typical errors that studentsmake?Pose an incorrect solutionAsk students to explain what went wrong.Sometimes show the incorrect process,sometimes just show the incorrect answer

Trainers might show this slide while participants are sharing Error Analysis problems they created.




Trainers can use the following: 1 - if you have an audience of mostly

geometry teachers 2 - if you want more examples using a

different strand



Slide 32 Another Example

The following slides begin with a differentstem problem based on geometry andscale factors.

Slide 33 A Typical Textbook Item

The length of a rectangle is 8cm and the width is 6 cm. Findthe perimeter of the newrectangle created by when theoriginal width is dilated by ascale factor of 4.

A typical item: Use the given information to find the new perimeter. Do students need to be able to do this? Of course. What other kinds of questions could teachers ask so that students learn more about scale factors and the relationships with surface area and volume?


as an Un-Doing problem?



If the volume is increased by a factor of8, what is the change in the length of theside of a cube?

Have participants briefly discuss this problem with a partner or group. In mathematics, we often do something and then un-do it. We multiply binomials, we factor the product. We add, we subtract. Here, instead of giving students the original information and having them use the scale factor to find the new dimensions and then the volume, students are given information about the volume and asked to go backwards to find the length. Do they understand the relationship



between the volume and the side length or have they just memorized a set of steps to do (to find the volume)?


Below is Craig’s work. What might havebeen the question?

16

3

34

8+6=144x3=12

Here is another “Un-Doing” example. Have participants briefly discuss this example, and ask them to consider how a student might think differently to solve this problem. Did this and the previous example seem more open? Might this possibly allow the teacher to see a greater variety of solutions and strategies that when discussed can build strength in connecting the different approaches? If students can do it one way, often it is a good assessment to see if they can back up, un-do the process, start with the answer and work back to find the problem.


as a snap shot problem?



Yesterday in classwe exploredfigures formed bydilations. Abbydropped herpaper in a puddle.Help her fill in themissing titles andvalues.

22

332 4 96 64

216

2

Another way to open up the discussion is to take a procedure or process and hide carefully selected parts of it. In this way students have to think about someone else’s strategies. This example also has an Un-Doing part to it. The third row asks them to find the scale factor given the original and new side length.




as an error analysis problem?



The length of rectangleOLDR was enlarged by ascale factor of 3 to createrectangle NEWS. If OLDRhas a width of 3 cm andperimeter of 16 cm, what isthe area of NEWS?Sandi wrote the following.What do you say to her?

35

L

DR

O

9

15E

WS

N

So, A= 9x15 = 90+45 = 135 cm 2

Here is a commonly made error. Both the length and the width were mistakenly enlarged instead of just the width as instructed. In this type of problem, students are asked to examine erroneous processes and find the error(s). Ask participants to consider the common errors of their own students. Suggest that instead of only re-teaching the correct method, they might also consider asking students to analyze the common errors made in their own classrooms.


a. 1/3

b. 1/2

c. 2/7

d. 5

What scale factorwas used totransform ² MNP to² RST?

Here is a released item from the 2004 9th grade TAKS. Ask participants to consider the various items at which they have just looked, and how they might have prepared students for this item and the next two. Note that this TAKS item asks students to find the scale factor given the figure’s dimensions. This is an un-doing question. Trainers might compare the original, typical item’s place on the pyramid with the locations of the other items.




a. The length is 2 times the original length.

b. The length is 4 times the original length.

c. The length is 6 times the original length.

d. The length is 8 times the original length.

If the surface area of a cube is increased by afactor of 4, what is the change in the length of thesides of the cube?

Here is a released item from the 2004 11th grade TAKS. Ask participants to consider the various items at which they have just looked, and how they might have prepared students for this item and the next. Trainers might compare the original, typical item’s place on the pyramid with the locations of the other items.


A rectangle has a length of 4 feet anda perimeter of 14 feet. What is theperimeter of a similar rectangle with awidth of 9 feet?a. 36 ft.

b. 42 ft.

c. 108 ft.

d. 126 ft.


Handout 1-1 What’s Your Problem? 6-25

Sample Assessment Items – Roger Throwing a Ball Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

1. Match the graph with the appropriate table. 2. What are the roots, solutions shown? ____________ 3. When will Roger and the baseball be at the same height?

_______ 4. Equations have _______ or ________, whereas functions

have _______ or _______. 5. What is Roger’s new starting height? ______

___a.

i.

___b.

ii.

___c.

iii



Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, Raul and Marco answered the following question. Who is right and why? 1. Given the table and graph representing Roger throwing a

baseball upward from a downward moving elevator, what are the roots, solutions shown?

Raul

Roots and solutions are the same as zeros. Y1 is zero at x = 0 and in between x = 3 and x = 4, so the roots are x = 0 and x is about 3.5

Marco

A root is the solution to an equation so we are looking for where Y1 = Y2. In the table, both Y1 and Y2 have the same values at x = 0 and x = 4. On the graph, Y1 and Y2 intersect at these same x values x = 0 and x = 4.

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What are 2 possible sets of lists that would produce the following graph:

Fill in the lists and note your window. Possibility 1: Possibility 2:



Rectangle RECT has an area of 4 cm2. It is enlarged to make the rectangle BIGS. BIGS has an area of 16. What was the scale factor used to enlarge RECT to get BIGS?



Sketch the view of a cube below, but make it appear three-dimensional by making some of the lines dashed.



Square SQAR has side length s. Choose your favorite scale factor to dilate SQAR to a similar square and label new square HUGE. Label the side lengths of HUGE in terms of s and find the area of HUGE in terms of s.



Write two quadratic functions with zeros x = 2, x = -3



A cube, P, has volume 1000 cm3. It is dilated by a scale factor of 0.5 to form a similar cube, S. What is the side length of cube S?



The clear cube shown has the letters DOT printed on one face. When a light is shined on that face, the image of DOT appears on the opposite face. The image of DOT on the opposite face is then painted. Copy the net of the cube and sketch the painted image of the word, DOT, on the correct square and in the correct position.

Discovering Geometry



The cube has designs on three faces. When unfolded, which figure at right could it become?




Find the equations of two linear functions that intersect at the point (2, 3)



Find the equations of two linear functions that do not intersect.



Write your two favorite linear functions with a y-intercept of 4



Write your two favorite linear functions with a slope of 4.



The green prism below right was built from the two solids below left. Copy the figure on the right onto isometric dot paper and shade in one of the two pieces to show how the complete figure was created.




Sketch the solid shown, but with the two blue cubes removed and the red cube moved to cover the visible face of the green cube.




Given f x( )= x2 + 3, find f 2( ) Cameron got the following answer. What do you think he did? What should he have done?

f 2( )= 2x2 + 6



The width of rectangle DAWN was enlarged by a scale factor of 2:3 to form a new rectangle RISE. What is the perimeter of RISE?

Justin’s work is below. What do you say to Justin?

10 to 15, 20 to 30 so 2(15) + 2 (30) = 30 + 60 = 90

90 cm

10 cm

20 cm D A

W N



Sanna solved the equation below as follows. Is she correct? If not, explain to her why.

x2 + 9x − 36 = 0x2 = −9x + 36

x2 = −9x + 36x = −3, 6



José answered the following question below. How do you respond to his answer? Include the correct response.

Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

What are the roots, solutions shown? _______

“The graph intersects the x-axis around 4.5, so the root (solution) is x = 4.5.”



For the figure shown, Kamisha drew the following orthographic drawing. How would you help her fix it?

Front Side Top



Lakrea created the net below. She is convinced that it will fold into a cube. Will it? If not, how would you convince her otherwise?



McKay says that the scatter plot below shows positive correlation because “the data goes up” as he pointed up from left to right. What do you say to McKay?



Bo was absent yesterday. When you started telling him about the Roger-throwing-the-ball-elevator activity, he said, “What’s the big deal? Roots, zeros, solutions, x-intercepts – they are all the same thing.” How do you respond to Bo? A complete answer includes graphs, tables, equations and discussion.



Yesterday we added marbles to cups hanging by rubber bands. We measured 3 distances, graphed the data, and found trend lines. Two groups results are below.

Group A

Group B

1. What can you conclude about Group 1’s marbles and rubber band compared to that of Group 2?

2. What can you conclude about Group 1’s table and cup compared to that of Group 2?



Yesterday we graphed different classes heights and arm spans. Two different classes are represented below.

Group A

Group B

1. Name one difference between the people measured by Group A and the people measured by Group B and explain how you know.

2. Name another difference between the people measured by Group A and the people measured by Group B and explain how you know.



Marisol spilled soda on her homework. Fill in the two missing steps covered by the stain.



For a week we have been bouncing balls under motion detectors and finding function rules to model one complete bounce. Here is some similar data. What function rule would you write to model the first complete bounce (the one the trace cursor is on)?



If the ratio of the heights in two similar figures is m/n, the ratio of their perimeters would be ______________, the ratio of their surface area would be _________________ and the ratio of their volume would be ____________________.



One of the following solids cannot be represented by the orthographic drawings. Change the orthographic drawings so that they represent the misfit and not the others.

top front side

a

b

c

d


Handout 4 What’s Your Problem? 6-58

Three Problem Types – How to Write

Snap Shot Problems: What are two ideas, processes, or representations that students mix up? Juxtapose them and ask which is which. What part of a large activity can you grab to assess if students got the gist of the large activity? Un-Doing Problems: Can you start with the answer? Can you start in the middle? Can you change one constraint? Can you start with a different representation? Ask students to create or invent the beginning of a problem. Error Analysis What are the typical errors that students make? Pose an incorrect solution. Ask students to explain what went wrong. Sometimes show the incorrect process; sometimes just show the incorrect answer.


The Power of Creating 6-59

Activity: The Power of Creating Overview: Teachers explore the power of creating problems. Materials: Two places to record answers (chart paper or white board space), each

with a coordinate axes and space for function rules Handout 1-Sample Class Responses, optional for use by Trainer

(page 6-63) Transparency 1-Slope of 4, optional one per group of 4 participants

(page 6-64) Transparency 2-Y Intercept of 4, optional one per group of 4 participants

(page 6-65) PowerPoint: The Power of Creating

Grouping: Partners Time: 30 minutes Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation The Power of Creating. Use the following note pages to elaborate on the content of each slide.


1

The Power of Creating

What do you mean?

In the set of problems that the participants classified as Un-Doing problems, there was a subset of those assessment items that are uniquely designed to teach while assessing. This subset is referred to here as “Creating Problems” because the students are asked to create or generate a response that contributes to a space of responses that as a whole increase students’ depth of understanding.

Slide 2 What’s Your Line?

Sketch a graph of your favorite line with ay-intercept of 4 and write the function rule.Sketch a graph of your favorite line with aslope of 4 and write the function rule.Post your graphs under the correspondingheadings and write the function rules.

Explain the 3 steps. Participants should work together in pairs to sketch the graph and write the function rule. Have one partner post the y-intercept graph and function rule, and the other partner post the slope graph and function rule in the appropriate places. As Trainers circulate, they should encourage some groups to be clever in their



choices. For instance, ask groups to consider negative numbers and all four quadrants. Ask some to use “complicated” numbers (fractions, decimals, really large, really small, etc.) On the board or on chart paper, designate two locations: one labeled “y-intercept of 4” and the other “slope of 4”. Or use the transparencies (pages 6-64 & 6-65) – one per group. Each group fills out one line with a function and graphs the same function. Then overlay them on the overhead projector.

Slide 3 Y-intercept and Slope

What do you see?

With both charts next to each other, discuss. Ask: What do you think?

What can you tell me about these two charts?”

Have participants focus on strategies. Ask: How did you find your line?

What were you thinking? Did anyone have a specific strategy

that you think will work every time?

Did anyone guess and check, using a graphing calculator?

Would your strategy work if I had asked for a y-intercept of 14 instead of a y-intercept of 4?”

Have participants share their strategies. Focus on the “y-intercept of 4” chart and ask the following questions if they did not already come up.

Ask: What can you tell me about these lines?” (They all intersect the y-axis at 4, the function rules all have a +4; they have different slopes; the function rules all have x1 (unless someone



chose y=4); they are all lines, …)

Focus on the “slope of 4” chart and ask the following questions if they did not already come up.

Ask: What do these sets of numbers have in common?” (They all have a middle number that is 5; they do not all have the same sum, …)

Slide 4 Summative

Find three equations of lines where theslope is -2 and three equations of lineswhere the y-intercept is -2. Graph thelines and label them with the functionrules.

Ask participants to find 3 equations of lines where the slope is –2 and 3 equations of different lines where the y-intercept is –2. Have them graph the lines and label them. This is a summative kind of task.

Slide 5 The Power of Creating

Powerful by themselvesMore powerful when put together to lookat commonalities.

Discussion: Creating Problems are powerful by themselves but even more powerful when the generated answers are compared and commonalities are found. This is a participant discussion. Get out of the content and broaden the discussion to Creating Problems in general.

Slide 6 The Power of Creating

Generative design centers ontaking tasks that typically convergeto one outcome and turning theminto tasks where students cancreate a space of responses.

Stroup, Ares &Stroup, Ares & Hurford Hurford, 2005, 2005

Ask participants to comment on this description of Creating Problems. What is a “space of responses”? How does this “space of responses” enhance learning?



Slide 7 Open Ended Questions

http://books. heinemann .com/math/

If your participants need support in using open ended questions, this sight has excellent resources that are free for teachers.

Slide 8 Rubrics

http://www. mathbenchmarks .org/rubric. htm

Grading help can be found at the Region 4 site: http://www.mathbenchmarks.org/rubric.htm This is the student rubric for grades 6-8.

Slide 9 Rubrics

http://www. mathbenchmarks .org/rubric. htm

This is the more detailed teacher rubric.

Slide 10 Other Examples:

Create the equation and graph of aquadratic function with y-intercept of 4and then again with a vertical stretchfactor of 4.Create a system of linear equationswhose solution is (3,2).

These are just two examples of Creating Problems. Ask participants to brainstorm with others and share. This same kind of activity could be done with quadratics. Find a quadratic with zeros x= 2, 4. What do they all have in common? What about all of the quadratics with zeros x= -2, -4? What about quadratics with y-intercepts of 2? What do their function rules have in common?


Handout 1 The Power of Creating 6-63

Sample Class Responses

y-intercept of 4 Function Rule Graph

1. y = 4 2. y = 4 − x 3. y = 4 + x 4. y = 3x − 4 5. y = −4 + 3x 6. y = 4 − 9x 7. y = 4 + 50x 8. y = 4 − 0.2x 9. y = 4 + 0.4x 10. 11. 12.

Slope of 4

Function Rule Graph 1. y = 4x 2. y = 2 + 4x 3. y = −2 + 4x 4. y = −15 + 4x 5. y = 12 + 4x 6. y = −8 + 4x 7. y = 7 + 4x 8. y = 0.5 + 4x 9. y = 18 + 4x 10. 11. 12.

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Closing Activity – Assessment Should Drive Instruction 6-66

Activity: Closing Activity – Assessment Should Drive Instruction Overview: Participants will discuss common definitions of Diagnostic, Formative, and

Summative assessment. Formative assessment, being defined as assessment that informs instruction, should dominate and be an integral part of mathematics classrooms. Several definitions are provided for discussion. Participants end the section by participating in an assessment scavenger hunt, looking for assessment items that appear all over the Assessment Pyramid.

Materials: PowerPoint: Closer

Current textbook(s) or local curriculum, optional Grouping: Tables of 4 Time: 30 minutes Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation Closer. Use the following notes pages to elaborate on the content of each slide.


1

Assessment

Mathematics TEKSRefinement Project

Briefly recall the Assessment Pyramid and the guiding questions. Ask participants to reflect on their current assessment practices. How do they use assessment to inform instruction?

Slide 2 Assessment

Ask participants to reflect on the assessment items they have seen in the presentation and to reflect on these guiding questions. • Do you have a clearer picture of the

difference between levels of reasoning and levels of difficulty, and how they interact?

• What items exemplify the kind of question for which students cannot just memorize their way through?



• What items exemplify the kind of question from which students can learn while answering?

Slide 3 Guiding Questions

How can I ask questions for which students can not justmemorize their way through? How can I ask questionsthat demand that students actually understand what isgoing on?How can I ask questions that students can learn fromwhile answering?How can I make sure that I have higher level reasoningquestions and not just more computationally difficultquestions?

Discuss the descriptions of diagnostic, formative, and summative assessment. Whichever it is called, assessment should be a routine part of the classroom, not an interruption. Formative – assessment that helps teachers make decisions about the content or form of instruction. Summative – used to judge students’ attainment. (Principles and Standards, p.24) “Some identify classroom assessment with formative assessment. We agree with Biggs (1998) that formative assessment and summative assessment are not mutually exclusive, as suggested by Black and Wiliam (1998). Their argument is that feedback concerning the gap between what is and what should be is regarded as formative only when comparison of actual and reference levels yields information that is then used to alter the gap. But if the information cannot lead to appropriate action, then it is not formative. Summative assessment in the form of end-of-year tests gives teachers the proof of how well they handled the formative assessment, assuming that the underlying philosophy is coherent and consequent. The differences in formative and summative assessment within the classroom are more related to timing and the amount of cumulation than anything else. Needed for both, of course, is that the assessment is criterion-referenced, incorporating the curriculum and resulting in aligned assessment.” (DeLange “Framework”, p. 4)



Slide 4 Assessment

Before the lesson (diagnostic)assessmentDuring the lesson (formative)assessmentAfter the lesson (summative)assessment

Use the rest of the slides to clarify any terms as they come up in the discussion.

Slide 5 Formative

Assessment should be more than merely a test at the endof instruction to see how students perform under specialconditions; rather, it should be an integral part ofinstruction that informs and guides teachers as they makeinstructional decisions. Assessment should not merely bedone to students; rather, it should also be done forstudents, to guide and enhance their learning (TheAssessment Principle, ¶ 1).


Slide 6 Formative

When the results of those activities are used inthis way—to adapt the teaching and learningpractice—we speak of formative classroomassessment

deLange

Slide 7 Formative

"When the cook tastes the soup, that'sformative assessment; when the customertastes the soup, that's summativeassessment."

Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment inEducation, 5, 7-74.



Slide 8 Summative

Judging students’ progress/ attainment

Slide 9 Degrees of Openess

Closed Task - one correct answer, oneroute to arriving at that answerOpen-Middled Task - one correct answerbut many routes to arriving at that answer.Open-Ended Task - several correctanswers and many routes to arriving atthose answers.

Slide 10 Assessing Mathematical Skills

“An assessment task that focus primarily onmathematical skills gives students a chanceto apply a well-practiced and importantprocedure or algorithm.”

Mathematics Assessment: A Practical Handbook, NCTM, 2000

Slide 11 Assessing Mathematical Skills

“These tasks are usually-routine;short;based upon recalling a well-known procedure;cast in a simple context or no context at all;focused on a single correct answer.”



Slide 12 Assessing Conceptual Understanding

“Assessment tasks that focus primarily onmathematical concepts give students achance to apply a concept in a newsituation, to reformulate it, and to express itin their own terms. These tasks probe theunderstanding of an idea.”


Slide 13 Assessing Conceptual Understanding

“They are usually-non-routine;short;based upon reconstruction, rather thanmemorizationcast in a context;focused on representation and explanation ofthe solution.”

Slide 14 Assessing Problem Solving

“An assessment task that focuses primarilyon mathematical problem solving givesstudents a chance to select and useproblem-solving strategies.”


Slide 15 Assessing Problem Solving

“Problem solving tasks are usually-non-routine;long;Predicated on the high-level use of facts,concepts, and skillscast in a context;focused on the students’ abilities to developand use strategies to solve.”






How has this presentation affected your view of assessment?

Slide 17 Scavenger Hunt

Using your current text and assessmentsfind examples all over the pyramid,justify your examples,share with the group.


Resources 6-72

Resources deLange, Jan. FRAMEWORK FOR CLASSROOM ASSESSMENT IN MATHEMATICS,

Freudenthal Institute & National Center for Improving Student Learning and Achievement in Mathematics and Science, September 1999.

National Council of Teachers of Mathematics (NCTM). Mathematics Assessment: A

Practical handbook For Grades 6-8. Reston, Va.: NCTM, 2000. National Council of Teachers of Mathematics (NCTM). Mathematics Assessment: A

Practical handbook For Grades 9-12. Reston, Va.: NCTM, 2000. National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation

Standards for School Mathematics. Reston, Va.: NCTM, 1989. National Council of Teachers of Mathematics (NCTM).. Principles and Standards for

School Mathematics. Reston, Va.: NCTM, 2000. Romberg, Thomas, A. ed., Standards-Based Mathematics Assessment in Middle

School: Rethinking Classroom Practice; New York: Teachers College Press. 2004. Serra, Michael, Discovering Geometry: An Inductive Approach. Emeryville, CA, Key

Curriculum Press. 2003. Stroup, W., Ares, N., & Hurford, A., (2005). A Dialectic Analysis of Generativity: Issues

of Network Supported Design in Mathematics and Science. Journal of Mathematical Thinking and Learning,7(3), 181–206.

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