Structures of Proportionality Problems Framework
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This document summarizes research on the structures of proportionality problems and how those structures impact student performance. It finds that problems become more difficult for students when (1) the multiplicative relationships are non-integral rather than integral, and (2) when both relationships are non-integral. Context, types of problems, and meaning of quantities also influence difficulty level. The document provides examples and case studies to illustrate these findings.
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Structures of Proportionality Problems Framework
- 1. Modified October 2008. Original materials created as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002) Structures of Proportionality Problems Krisan Stone, VMP Leslie Ercole, VMP Marge Petit, Marge Petit Consulting (MPC)
- 2. OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002) Mathematical Topics And Contexts Structures of Problems Other Structures Evidence in Student Work to Inform Instruction Proportional Strategies Transitional Proportional Strategies Non-proportional Reasoning Underlying Issues, Errors, Misconceptions
- 14. OGAP Study Findings (2006 Pilot, n=153) October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002) Multiplicative Relationships within and between figures Percent of Correct Responses Pilot 1 Both integral 80% Pilot 2 One integral, one non-integral 65% Pilot 3 Both non-integral 35.5%
- 31. Structures of the Problems OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002) Old Playground 90 ft . 630 ft. A school is enlarging its playground. The dimensions of the new playground are proportional to the old playground. What is the measurement of the missing length of the new playground? Show your work. What type of problem is this similarity problem? New Playground 110 ft.
- 32. Structures of the Problems October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002) OGAP Proportionality Framework The dimension of 4 rectangles are given below. Which two rectangles are similar? 2” x 8” 4” x 10” 6” x 12” 6” x 15” What type of problem is this similarity problem?
- 33. Structures of the Problems October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002) OGAP Proportionality Framework Jack built a scale model of the John Hancock Center. His model was 2.25 feet tall. The John Hancock Center in Chicago is 1476 feet tall. How many feet of the real building does one foot on the scale model represent? Be sure to show all of your work. The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? What is the general structure of scale factor problems?
Notes de l'éditeur
- 2:00-2:10 (slides 1-8) Our plan is to share with you our thinking in designing content sessions for the project. We will talk about our goals--goals are based on what we considered to be the central challenge in teaching. We will analyze a couple of tasks for their potential to support student learning. We will describe a framework for thinking about the the interaction of task with the factors that influence its implementation. Lastly we will give you the opportunity solve a high-level task and experience the benefits of analyzing various solutions. Something about making this as concise as possible, but sufficiently elaborate to provide a good sense of the work in which they will be engaged during the course of the project. Mention TIMMS & QUASAR
- Slide 18: 1 min The primary research that underlies OGAP proportionality is --- see statement on the slide. OGAP professional development is designed to develop the understanding of the interaction between the structures of problems, contexts in which proportionality problems are situated, and the strength of proportional reasoning, and the resulting instructional implications. 615/09 Research shows that student’s use of proportional strategies varies with problem contexts.
- Slides 2-9, 25 min. Research indicates and VMP pilots support that there are primarily 5 structures that influence the strategies students use to solve problems involving proportionality The multiplicative relationship in a problem Context of problem Types of problems The meaning of quantities Complexity of the numbers In this session we will be focusing on --- The multiplicative relationship in a problem Context of problem Types of problems Meaning of the quantities Skipping complexity of numbers--pretty obvious.
- Slides 2-9, 25 min. NOTE: In most cases participants have already identified this structure in the Structures Case Study – not necessarily by stating it in this way – but when they placed problem number 1 (Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?) in the “Easiest” category. It usually is placed there for two reasons – small numbers – and – in participant words – “It is easy to see that 8 is 4 x 2.”
- Slides 2-9, 25 min. The first thing that needs to be established is what is meant by “within” and “between” ratios. The slide illustrates this idea, BUT we have found that we must continuously have teachers think and rethink this idea. In this case the multiplicative relationship “within the ratio (boxes to bushels)” is non-integral (1.5x), while the multiplicative relationship “between the ratios (4x)” is integral (4 x). Background: Some of the literature refers to “within” and “between” measure spaces, not ratios. For example – “within” bushels as one measure space or “between” boxes and bushels as two different measure spaces. However, this did not resonate with the field. We were advised to use “within” and “between” ratios which has resonated with teachers.
- Slides 2-9, 25 min. The first thing that needs to be established is what is meant by “within” and “between” ratios. The slide illustrates this idea, BUT we have found that we must continuously have teachers think and rethink this idea. In this case the multiplicative relationship “within the ratio (boxes to bushels)” is non-integral (1.5x), while the multiplicative relationship “between the ratios (4x)” is integral (4 x). Background: Some of the literature refers to “within” and “between” measure spaces, not ratios. For example – “within” bushels as one measure space or “between” boxes and bushels as two different measure spaces. However, this did not resonate with the field. We were advised to use “within” and “between” ratios which has resonated with teachers.
- Slides 2-9, 25 min. Have participants write the ratios two different ways and then identify the multiplicative relationship “within” and “between” the ratios. Multiplicative relationships within and between for this problem. 3 boxes: 2 bushels = x boxes:7 bushels (“within” – non-integral (1.5 x), “between” – 3.5 x) 3 boxes: x boxes = 7 bushels: 2 bushels (“within” – non-integral (3.5 x), “between” – 1.5 x)
- Slides 2-9, 25 min. The most difficult situation is when the multiplicative relationship is non-integral both “within” and “between” the ratios.
- Slides 10-14, 25 min. Segue to first component of the Framework.
- Slides 10-14, 25 min. Background: To study the impact of the multiplicative relationship within and between ratios the Vermont Mathematics Partnership conducted a pilot study involving 153 seventh grade students. In the pilot study students solved three versions of the same problem at three different times across one week. The main difference between the three problems was the nature of the multiplicative relationships within and between figures. Pilot 1: A school is enlarging its playground. The dimensions of the new playground are proportional to the dimensions of the old playground. What is the length of the new playground? Old Playground New Playground
- Slides 10-14, 25 min. This activity is divided into two parts and is designed to engage participants in the impact in student work that results when the multiplicative relationships “within” and “between” ratios is altered. Part 1 is focused on the problems used in the study. Part 2 is focused on an analysis of student work. Part 1: Hand out the Case Study to participants Follow the directions on the slide. Before going onto to Part 2, engage in discussion with participants. This is an opportunity to assure that participants understand what is meant by – “within” and “between” the ratios. Part 2: Hand out student work sets. Explain that the student work is organized by students and they should be reviewed that way. (e.g., review student 1 – pilot 1, then pilot 2, and then pilot 3. Record the data for student 1 as you review the work. Then go to student 2.) Follow the directions on the Case and the corresponding PP slide. Before reviewing the data from the Vermont Mathematics Partnership pilot study on slide 13, engage in a discussion with participants relative to discussion questions on the Case and on slide 12.
- Slides 10-14, 25 min. Before conducting a group discussion it is recommended that the pairs share their responses to the questions with the rest of their table. What did you see that you expected? [Most participants will say that they expected better results with problems 1 then with problems 2 and 3.] 2) What surprised you? [Most participants will be surprised by a couple of things. 1) that students can use a multiplicative strategy for problems 1 and 2, but revert to additive for problem 3; 2) students will stay with a “between” or “within” strategy even when one has an integral relationship and their choice doesn’t.] 3) What are implications for instruction and assessment? [This is the most important question. Participants usually get a big “aha’ at this point (or start t0). That is, they realize that they need to pay attention to assuring that students have experience solving problems involving proportionality that vary the multiplicative relationship “within” and “between” ratios. As well as not assume that if students get an 80% on an assessment focused on proportionality, that they are proficient without paying close attention to how students handle problems with varied multiplicative relationships “within” and “between” the ratios.]
- Slides 10-14, 25 min. The results of the VMP Pilot Study supported findings by other researchers.
- Slides 15-24, 15 min. Segue to the Context feature of the Framework
- Slides 15-24, 15 min. The next two slides are designed to call attention to these two points, NOT by providing examples of all the possible contexts, but by showing three different contexts and three different ways in which proportionality “shows up” in problems. Have students take out their OGAP Proportionality Framework. Also, be ready to refer to the graphs made by participants in the Proportionality Activity. ** Pat: We were not sure about the wording on this slide.
- Slides 15-24, 15 min. The most familiar context is Nate’s shower and the least familiar is the scale factor problem. Proportionality shows up in three different ways in these problems. Rectangle problem: Using the scale factor to scale down Nate’s shower problem: Application of a unit rate Toasty Oats problem: Rate comparison
- Slides 15-24, 15 min. Segue to “Types of problems”
- Slides 15-24, 15 min. Please take out the OGAP Proportionality Framework. Locate the “Problem Types” on the framework. As we review each of these types we will look closely at the structures within the problems.
- Slides 15-24, 15 min. There is often lack of agreement between the definitions of ratios and rates. There are some instances where a case can be made either way.???????????????? For the sake of a common definitions for this discussion these definitions have been adopted. This slide and the examples that follow are meant to exemplify these definitions.. The “big idea” is that ratios are comparisons of like quantities – people to people OR eyes to eyes, while rates compares two different quantities and describes how one quantity depends upon the other quantity. [New-we find a lack of agreement about these definitions. For the sake of having common definitions, we are using ones that are frequently found in middle grades texts. We will find that there are situations for which an argument can be made for categorizing a comparison as either a ratio or a rate. In these cases, we find it is not productive to pursue the issue.]
- Slides 15-24, 15 min. Refer to the Framework: You will notice two structures related to ratios on the framework: Ratio Relationships and Ratio Referents. The relationships in ratios can be part to part OR part to whole. In addition, the reference to the whole or part may be explicitly stated or implied in the problem. Let’s look at a couple of examples on this slide and the next slide.. Read the problem. With a partner answer the following questions. Is the relationship a part to part OR a part to whole relationship? [This is a part to whole – 7 th grades boys (part): 7 th grade students (whole)] Is the whole explicitly given or implied in the problem and data given? [The whole is implied. What is given is the two parts – number of girls and number of boys – not the whole – the 7 th grade students.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
- Slides 15-24, 15 min. Read the problem. With a partner answer the following questions. Is the relationship a part to part OR a part to whole relationship?[This is a part to part problem – red marbles: blue marbles.] Is the whole explicitly given or implied in the problem and data given? [The parts are explicitly stated.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
- Slides 15-24, 15 min. As mentioned earlier rates are comparisons of two different quantities where one quantity is dependent on the other quantity. Rate problems assume you start with two different quantities and end with an entirely different type of quantity. For example, this problem provides a rate (speed as defined by miles per hour), the time (in hours), and asks for a distance (undefined). Instructionally it becomes important for students to think about the meaning of the quantities, not just the units. One way to help students focus on the meaning of the quantities is to have students model the situation. Review the model that represents the situation. Explain how this model illustrates the meaning of 55 miles per hour. [Note: teachers claim that many students think miles per hour is one word – milesperhour. This model helps students to understand that miles per hour means miles per every hour.]
- Slides 15-24, 15 min. Have participants work in pairs to analyze these two rate comparison problem. [In general, in rate comparison problems the two quantities that make up the rate are given, but not the rate to be compared.] NOTE: Some participants might say that these problems are built to “trick” students because the order in which the quantities (e.g., horses and acres) are given in the problem statement is not the same as the rates students are asked to compare (e.g., acres per horse). If this occurs, please point out that OGAP questions are designed to elicit fragile understandings – they are formative, not summative. You want to know if your students are paying attention to the quantities, so questions are designed to determine if students are attending to the problem situation.]
- Slides 25 & 26, 15 min. TAB 5 => MEANING OF QUANTITIES CASE STUDY To explore the importance of placing an instructional focus on “meaning of the quantities” in problems complete the “Meaning of Quantities” Case Study. Hand out the Meaning of the Quantities Case Study materials. Use this slide to introduce Part I of the Case Study. Provide about 10 – 15 minutes for participants to work in pairs. Then about 10 minutes to debrief the activity focusing on general instructional strategies that arise from the group. [See Facilitator Notes for an analysis of each of the solutions.]
- Slides 25 & 26, 15 min. TAB 5 => MEANING OF QUANTITIES CASE STUDY To explore the importance of placing an instructional focus on “meaning of the quantities” in problems complete the “Meaning of Quantities” Case Study. Hand out the Meaning of the Quantities Case Study materials. Use this slide to introduce Part I of the Case Study. Provide about 10 – 15 minutes for participants to work in pairs. Then about 10 minutes to debrief the activity focusing on general instructional strategies that arise from the group. [See Facilitator Notes for an analysis of each of the solutions.]
- Slides 25 & 26, 15 min. Individually and then as a group analyze the student solution in Part II. [See Facilitator Notes for the Case for an analysis.]
- Slides 27-30, 10 min. Have participants take out their OGAP Framework. Please take out the OGAP Framework. The next Problem Type that we will analyze for structures is missing value problems. What is the general structure of a missing value problem? [In general, missing value problems involve finding a missing value in a set of equal ratios. That is, three of the four quantities are given and the solution involves finding the fourth quantity.] These should go quickly.
- Slides 27-30, 10 min. Please refer to the OGAP Framework and find “Internal Structure.” You’ll notice a reference to Internal Structures. In the Ranch problem many of you noticed that the quantities were given in one order (acres and horse), but the rate was asked for in a different order (horses per acre). This is an example of non-parallel structure. In missing value problems researchers also suggest that the location of the missing value matters. The implications for instruction and assessment is that the location of the missing value should be varied in problems that students solve. Review of problems: The quantities in both these problems are the same. In both problems the solution is the number of boxes. However, These two problems are the same except that the missing values are not in the same place in the problem statement. In the top problem the structure is parallel (i.e., boxes to bushels throughout). In the second problem the structure is NOT parallel. The first ratio is given as boxes to bushels. Then the number of bushels is given. Then students are then asked to determine the number of boxes. .
- Slides 27-30, 10 min. [Change: As a whole group, briefly discuss how to modify this problem to be easier and harder. Share a few of the problems that participants write.] Have participants work in pairs for a few minutes to modify this problem to be easier, harder. Share a few of the problems that participants write. Things that participants will probably change: the order of the quantities, the magnitude of the numbers, or the multiplicative relationship within and between the ratios.
- Slides 27-30, 10 min. Similarity problems show up as missing value problems, ratio comparison problems (See slide 31), and scale factor problems (see slide 32). This is a missing value problems.
- Slide 31, 5 min. This is a ratio comparison problem. It can be solved by comparing the ratios between the two dimensions or by finding the multiplicative relationship between the dimensions. Ratio comparison solution (B and D because the ratio of one length to the other is the same (2:5)) 1:4 2: 5 1:2 2: 5 Multiplicative relationship between dimensions (B and D because the multiplicative relationship with both ratios is 2.5) 8” is 4 times 2” 10” is 2.5 times 4” 12” is 2 times 6” 15” is 2.5 times 6”
- Slide 32-33, 2 min. [Go fast] The examples illustrate two different structures for scale factor problems. John Hancock Center Problem: The height of the model and the original are given. The problem asks for scale factor. Rectangle: The scale factor is given with length of the larger rectangle. The problem asks for the length of smaller rectangle
- Slide 32-33, 2 min. In pairs, participants modify this problem. Ways to modify problem: Change scale factor to an integer Give dimension of smaller rectangle and ask to scale-up instead of scaling down
- Slides 34 & 35 (5 min.)
- Slides 34 & 35 (5 min.) These are non-numerical problems that involve a proportional situation. These help students think about the relationships, and not take cues from numbers.
- Slides 36-37 (15-20 min.)
- Slides 36-37 (15-20 min.) These two problems were given to a group of pre-service teachers. [If participants have not already solved Sue or Julie (or a problem like it) then have participants solve both these problems. Then re view solutions. It has been our experience that many teachers will apply a proportional strategy to the additive situation in the Sue and Julie problem. This is consistent with what happened with the group of pre-service teachers. See the next slide.] ** Slide 37 and 38 could be interchanged depending on your goals.
- Slides 38-41 (5 min.)
- Slides 38-41 (5 min.)
- Slides 38-41 (5 min.) TAB 6, Non-proportional Student Work Sorting Task To determine if middle school students treated this additive situation as a proportional situation, the VMP OGAP conducted a small study involving 82 sixth grade students. Some of the students had had instruction in solving proportional problems and others did not have instruction prior to solving the problem. Handout the Non-Proportional Student work set of papers. To get a feel for the type of responses found in the study (with a partner) sort this student work into 2 piles – 1) treated the problem as a proportion problem; 2) Treated as an additive problem. What did you find?
- These are the data from the study. What are the instructional implications of these data? Have participants discuss in pairs for a minute or two and then as a full group. [Usually the discussion starts with the realization that the results might be an artifact of instruction. That is, most mathematics programs/texts ONLY include proportional problems during units focused on proportions. Students may use the structure of 3 known quantities and one unknown quantity and assume it is a proportional situation. From there participants recognize the importance of embedding non-proportional problems in instruction and assessment so students have to discriminate between the situations.]
- The focus here should also be on the instructional implications assuring that students interact with a variety of structures in both instruction and assessment. Q: We want to verify that the whole slide, in particular the fourth bullet makes sense.
- JF & NC: We are not sure of the wording of the second bullet. (given a variety; regardless of the structure; or other suggestions?)
- Hand out the OGAP Framework and grade level pre-assessments. Regroup participants into grade level groups (if this makes sense this late in the workshop). Have participants analyze each of the problems for 1) problem types;2) Context; 3) multiplicative relationships within and between ratios; 4) Internal structures; and, 5) ratio relationships and referents
- Thoughts on Administering the OGAP Proportionality Pre-Assessment An important component of the Vermont Mathematics Partnership’s Ongoing Assessment Project involves gathering information about student understanding of proportionality concepts before beginning instruction through the administration of a pre-assessment. This pre-assessment is designed to elicit developing understandings, pre-conceptions, misconceptions, strategies that student use, and common errors that students make when solving questions involving proportionality. It is in this spirit of formative assessment that we offer the following thoughts on administering the pre-assessment. Tips for Students Let the students know that this is a pre-assessment on material that they will be learning this year so some or all of the material may be new to them. Encourage them to try their best even if they are unsure. Remind them that the information will help you in your planning, and will not be used as a grade. Time The amount of time students need to complete the pre-assessment will differ depending on the grade level and the number of items in your pre-assessment. The pre-assessment can be administered in numerous ways. Some teachers choose to spread the assessment over several days while others administer the entire assessment in one class period. Again, the purpose is to collect evidence from your students so feel free to choose a schedule that works best for your students. Level of Teacher Assistance The purpose of formative assessment is to collect evidence that will help you best meet the needs of your students. With this in mind, feel free to read any items to students who you feel need this type of accommodation. You may also decide to scribe for students who require assistance with writing. Although no special materials are needed to complete the pre-assessment, students can use tools or manipulatives that are part of regular classroom instruction. By all means assist students in decoding non-mathematical vocabulary. You should not, however, help students interpret any mathematics content. Final Thoughts The ideas above are not intended to be used as a “checklist of do’s and don’ts” but rather as a way to communicate the spirit in which the pre-assessments are best administered to your students. Please bring the completed pre-assessments to the December session. A major goal of these sessions is to help you learn how to analyze the evidence in your students’ responses and use your findings to influence your upcoming proportionality instruction. Feel free to contact Marge Petit ( [email_address] ) if you have any questions.