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Insights Into Algebra 1: Teaching for Learning

Quadratic Functions Teaching Strategies: Alternative Assessment

The educational community today is focusing more and more on assessment. High school exit examinations are being used to determine whether a student has attained the minimum level of education required to graduate. While correct in theory – students should meet certain minimum standards – large-scale standardized tests only provide a glimpse of a student’s ability.

The National Center for Fair & Open Testing, in its fact sheet “The Limits of Standardized Tests for Diagnosing and Assisting Student Learning,” states:

To improve learning and provide meaningful accountability, schools and districts cannot rely solely on standardized tests. The inherent limits of the instruments allow them only to generate information that is inadequate in both breadth and depth. Thus, states, districts and schools must find ways to strengthen classroom assessments and to use the information that comes from these richer measures to inform the public.

This problem, however, is not specific to standardized testing. In the classroom, evaluating students with only one type of assessment is ineffective. It is impossible to gain an accurate picture without allowing multiple opportunities for students to demonstrate what they know and can do.

To enable students to demonstrate the full scope of their mathematical understanding, assessment should include performance tasks, individual and group presentations, essays, class discussions, and research projects. Tony Piccolino notes how Tremain uses assessments that address the needs of all his students.

Transcript from Tony Piccolino

Tremain is very interested in making sure that he addresses the needs of the students by focusing on their different learning styles. Some students learn a mathematical concept graphically, some learn it better numerically, some analytically, and some need to verbalize it in order to make sense of the mathematics. And in this lesson, Tremain focuses on what they commonly call in math education the Rule of Four – getting students to represent a concept graphically, numerically, analytically, and verbally. All students should learn representations, but Tremain recognizes that each student has his or her own strength in one of those representations, and he makes very effective use of each of those in the lesson.

Using Rubrics

A rubric is a list of criteria that describes the range of possible responses to a mathematical task, from exceptional to unacceptable. Assessing a performance task with a rubric allows a teacher to assess a student’s learning thoroughly.

The key benefit of using a rubric to evaluate a performance task – as opposed to using a multiple choice test and an answer key – is that teachers are able to make informed determinations about the evidence a student provides. For instance, consider a question that asks students to identify the vertex of the parabola y = -x2 + 12x. On a multiple choice test, if the correct choice is, say, C, (6, 36), and a student selects B, (6, -36), the answer is clearly wrong. On the other hand, imagine that a student provides the following work for the same question:

Although there’s no way to know for sure, it appears that this student’s only mistake is incorrectly identifying the vertex from the equation written in vertex form. It could be a careless oversight, or it may be that the student doesn’t know how to determine the coordinates of the vertex using the vertex form. Although the latter indicates a conceptual error, the student has nonetheless demonstrated that he or she can write the equation of a quadratic in vertex form when given a graph. Yet on a multiple choice test, this student would receive the same amount of credit as a student who didn’t even attempt the problem.

As an example of how a teacher might use a rubric to evaluate a performance task, the above problem (finding the vertex of a parabola) has been converted to a performance task below. The accompanying rubric suggests the criteria for evaluating a student response.

Original Problem:
The vertex of the parabola y = -x2 + 12x occurs at:
A. (-12, 0)
B. (6, -36)
C. (6, 36)
D. (12, 0)

Performance Task:
The path of a model rocket can be described by the quadratic function y = -x2 – 12x, where the point (x, y) represents the height (y) of the rocket (in meters) at time x seconds after takeoff. Identify the maximum height of the rocket, and determine the time at which the rocket reached its maximum height.

The sample student work given above, in which a student effectively completed the square and transformed the equation to vertex form, would likely be evaluated as Level 3 according to this rubric – the student used a correct method, but a minor error prevented a correct solution.

The National Council of Teachers of Mathematics (NCTM) promotes the use of performance tasks and rubrics as a means of accurately assessing student ability. Further, NCTM believes that using rubrics may be one factor in assuring that students take responsibility for their own learning. In Principles and Standards for School Mathematics (PSSM), NCTM states:

Feedback from assessment tasks can also help students in setting goals, assuming responsibility for their own learning, and becoming more independent learners. For example, scoring guides, or rubrics, can help teachers analyze and describe students’ responses to complex tasks and determine students’ levels of proficiency. They can also help students understand the characteristics of a complete and correct response.

This last point – using scoring guides to help students understand the characteristics of an acceptable response – is illustrated in Tremain Nelson’s classroom. For some projects, students in Tremain’s class grade one another’s work using an assigned scale for each category. For other projects, the students themselves generate categories that will be used to assess their work.

Transcript from Tremain Nelson

Their grades during their presentations came from the set of criteria that came out of our table talk discussion. They had input, which gave them buy in. And then, during their presentation, they knew exactly what they needed to do. When they got up and did the presentation, they knew what I was grading for, because they decided that that’s what I should look for.

NCTM advocates activities of this sort, stating in PSSM:

Through the use of good tasks and the public discussion of criteria for good responses, teachers can cultivate in their students both the disposition and the capacity to engage in self-assessment and reflection on their own work and on ideas put forth by others. Such a focus on self-assessment and peer assessment has been found to have a positive impact on students’ learning (Wilson and Kenney forthcoming).

For another example of a performance task that teachers can use a rubric to grade, look at the Rectangular Plot activity and its rubric. The Rectangular Plot activity is a performance task that students can solve with a quadratic equation.
(Source: Danielson, Charlotte and Elizabeth Marquez. A Collection of Performance Tasks and Rubrics: High School Mathematics. Larchmont, NY: Eye on Education, 1998; p. 209.)

A Rectangular Plot
Suppose that you want to enclose a 650-square-meter rectangular plot of land along a river. What can the dimensions of the plot be if you have only 110 m of fencing and you do not fence the riverside? Provide a labeled diagram and show the method you used to arrive at your answer. If a graphing utility is used, indicate how it was used, and if applicable, sketch and label curves and any significant points. (Round answers to the nearest tenth.)

Rectangular Plot
Rubric

Level 4: This response offers clear and convincing evidence of a deep knowledge of the mathematics related to this task.

Characteristics:
Student correctly solves the problem by one of the methods given. [The equations resulting from the problem scenario are L + 2W = 110 and LW = 650. One possible method of solving this system is graphing a quadratic function in L or W and finding the roots of the equation – the equations are W2 – 55W + 325 = 0 and L2 – 110L + 1300 = 0.] Both solutions [48.3 m by 13.5 m, and 6.7 m by 96.5 m] are given. Work may have minor flaws – e.g., intersection not labeled, or one of the answers may not be correctly rounded.

Level 3: This response offers evidence of substantial knowledge of the mathematics related to this task.

Characteristics:
Student attempts to solves the problem by one of the methods given and both solutions are correct except for one incorrect or omitted dimension.

Level 2: This response offers limited or inconsistent evidence of knowledge of the mathematics related to this task.

Characteristics:
Student attempts to solve the problem by one of the methods given but states only one set of correct dimensions; or states only the widths and not the lengths of both sets of dimensions or vice versa; or the student states both lengths or both widths as one set of dimensions.

Level 1: This response offers little or no evidence of knowledge of the mathematics related to this task.

Characteristics:
Student attempts to solve the problem by one of the methods given but cannot correctly state any part of either set of dimensions, or only one dimension of one set is correct.

To fully understand rubric scoring, and to recognize how useful and easy it is to do, you may wish to apply a rubric to student work from your classroom. For example, you might score some student work using the general math rubric given in the Exemplars site listed in the Resources section, just to get a sense for rubric scoring. The additional links and references listed in Resources give examples of scoring rubrics, offer help in creating your own rubrics, and show performance assessments.

Reflection:

Consider a few performance tasks that you already use with your students. Think about how you evaluate student work during these activities. Explain how your method of assessment ensures that student work is graded fairly, and give examples to demonstrate that your students understand the criteria upon which their evaluations are based.

Team Presentations

Effective teachers would not expect students to give a four-person presentation during the first week of school. Building up to that takes time. Tremain Nelson, aware that students have to develop confidence in their ability to discuss mathematics, prepares students by slowly phasing in presentations as assessment activities throughout the year.

Listen to audio clip of teacher
Tremain Nelson

 

Transcript from Tremain Nelson

When I first began introducing presentations, and we do them very often in the class, I found that the students were nervous, they were shy, they did not necessarily want to go up and speak. It takes time, and toward the end of the year we are able to do a full-blown presentation. And there are different types of presentations that we can do. This type of presentation that you saw today, it was a presentation where the student had sole responsibility of everything and the teacher only sat back and listened. You don’t do that type of presentation from the beginning. From the beginning you may do a gallery-type presentation in which you place posters all around the room and you have students sitting, kind of like a science fair, and they walk around and they view each other’s presentation and you give them an opportunity to talk about what they’ve done. This can kind of get rid of that anxiety so that when they actually have to stand up and do the type of presentation you saw today, they’ll be ready.

Listen to audio clip of teacher educator
Tony Piccolino

Transcript from Tony Piccolino

The presentations by the students were a particularly effective way of promoting standards-based mathematics in the classroom. They help students develop the idea that this is a community of learners in which students help one another, they value each other’s ideas, and they also listen to other students for constructive criticism. It also helps to promote the view of Tremain as being a facilitator and not just imparting knowledge and information to the students. This notion of community of learners and the teacher as a facilitator is a very effective way to promote student understanding, student self-esteem, and confidence in learning mathematics.

Students in Tremain’s class progress from informally discussing mathematics in two-person teams (at the beginning of the year) to giving unscripted group presentations (by the end of the year).

Two-person Teams
At the beginning of the year, Tremain’s students solve problems in teams of two. Research shows that students who participate in group work experience increased learning (See R. E. Slavin, “Learning Together: Cooperative Groups and Peer Tutoring Produce Significant Academic Gains,” in American Educator, Summer 1986; pp. 6-13.) In addition, using smaller groups encourages students to participate. In a larger group, a student could easily remain silent while other members dominate; in groups of two, students are required to express their opinions in order to continue the conversation. When talking with a peer, a student has a chance to confront his or her mistakes. During group interaction, students may clarify thinking and resolve misconceptions; in addition, group work fosters social skills and develops a safe environment in which students can think about mathematics. By working in pairs from the beginning of the year, Tremain’s students become comfortable discussing mathematics and sharing their thoughts.

Gallery Presentations
Later in the year, Tremain has students exhibit their work in gallery presentations. This mode of presentation involves teams of students displaying their work while other students and the teacher engage them in informal conversations about it. Talking about mathematics in this way puts more responsibility on the individual, because a student has to be able to discuss his or her work intelligently. However, the environment of a gallery presentation is less threatening than a front-of-the-room presentation because a student is not the center of attention in a 30-person classroom. Instead, several teams present concurrently, and each team only has to present to one other team at a time. Then they switch roles, and the team that presented first gets to listen to the other team’s report. The teams rotate around the room so that everyone hears every presentation by the end of the period.

Scripted Presentations
Students in Tremain’s class then progress to preparing and giving scripted presentations. Students are given a framework for creating their presentations, which includes details on what to include in the introduction, which examples to use, how they should be presented, and so forth. While slightly more intense than gallery presentations, because the team has to present in front of the entire class, scripted presentations allow students to prepare all of their material in advance. There are no surprises, and students feel somewhat more at ease than if they had prepared the presentation on their own.

Transcript from Tremain Nelson

The first time they actually have to stand in front of a class and do a presentation – then what I do is I give them a scripted presentation so they know exactly what it is that they’ll have to say, there’s not a lot of fear involved. They know they are going to say the right thing; it’s just about getting the courage to stand up and do it.

Unscripted Group Presentations
Near the end of the year, as the final step in becoming confident public speakers, students prepare group math presentations entirely by themselves. Prior to the presentations, the class collectively generates a set of categories that outline the criteria for an excellent presentation. Using these as a guide, and working in teams of three, students decide every detail of their presentation: what material to cover, how to present the information, which examples to share, and which group member will present each piece.

Transcript from Tremain Nelson

While the groups are presenting, the class is taking notes in a special form. They are writing questions for the group for when they finish. They are writing positive comments for the group and they are giving them comments about what they would change or do better for the next time that they present. Now the purpose of that is actually twofold. One is that we want the students to be active listeners. We don’t want them to be paying attention to other things or thinking about their own presentation that they will have to do eventually. The other thing that’s important about that is that the feedback that they give to the presenter is going to help them, and it’s going to help the presenters.

Reflection:

Some teachers believe that presentations have no place in the mathematics classroom – they simply take too much time away from learning content. However, because a key component of the standards is for students to communicate mathematically, other teachers believe that presentations are absolutely vital. Do you believe that presentations are an effective and necessary form of assessment? Justify your position.

Peer Review

While some teachers are skeptical about relinquishing authority to students, researchers Paul Black and Dylan William believe that peer assessment is valuable and effective. “Pupils are generally honest and reliable in assessing both themselves and one another; they can even be too hard on themselves,” the pair writes in “Inside the Black Box: Raising Standards Through Classroom Assessment,” an October 1998 article in the Phi Delta Kappan. “The main problem is that pupils can assess themselves only when they have a sufficiently clear picture of the targets that their learning is meant to attain.”

In Tremain Nelson’s class, students use a teacher-generated list of categories to evaluate the work of other teams. In addition, the teacher and the students evaluate student presentations using a list of categories that are jointly designed. Consequently, students in Tremain’s class understand the targets to which their learning is supposed to be guided.

Listen to audio clip of teacher
Tremain Nelson

 

Transcript from Tremain Nelson

Having the students grade the work that they are seeing, that another student produced, does two things. First, it gives them an idea of what I would expect to see if I was grading. Secondly, it also gives them ownership of the activity. By asking them to actually grade it, that encouraged the students to actually go step by step and look at the thought process that someone else used and then determine whether or not that thought process is acceptable for that problem.

Transcript from Tony Piccolino

Tremain uses effective questioning techniques to help assess his students’ understanding of the mathematical concepts that are being taught. And he also asks other students to respond to any ideas developed by other students, so that he could assess a wider range of students. What I particularly liked about Tremain’s assessment is that not only does he assess students, but he also allows students the opportunity to assess one another in a real sense. They are doing a self assessment of their own learning, which is really very important.

If you’re not convinced that peer review could work in your classroom, try the following experiment: Give your students an assessment task to complete and grade the task using a rubric or another list of criteria. Then, allow your students to evaluate one another’s work using the same criteria. Compare the results of your evaluation with theirs. In addition to learning that their peers’ evaluations are typically “honest and reliable,” as Black and William suggest, your students might also notice some evidence of understanding that you were unable to detect.

Reflection:

In your classroom, what opportunities exist for students to evaluate their peers? What, if any, are the positive effects of allowing students to assess one another’s work?

Self-paced Technology

Computer technology has not yet reached a level where it removes the need for teachers – and, likely, it never will. But computers do have one advantage over teachers: they can provide immediate and specific feedback. Although the technology is not yet available everywhere, some schools have computers that allow students to understand the mistakes they are making in real time, and various software packages provide feedback to help them correct their misunderstandings.

In Tremain Nelson’s class, students use the computer every day. Working at their own pace, students progress through a computer-based math curriculum. The Carnegie Cognitive Tutor program is designed to incorporate class work and computer lab work. Students are expected to spend 40 percent of the total class time working independently on supplementary problems in the computer lab. The program monitors student progress and keeps track of their success. Students cannot pass to the next level in the program until they reach an acceptable level of competence in their current level. As students solve problems and learn content, the software checks for conceptual understanding. If students provide enough evidence to ensure that they understand, they move on to the next topic or lesson; if not, the software – and the teacher – offer remediation to clarify any misconceptions.

Listen to audio clip of teacher
Tremain Nelson

 

Transcript from Tremain Nelson

If you think about how you would help a child during a personal one-on-one human tutor, if that child didn’t have any difficulties at all, then you wouldn’t say very much to them. You would allow them to move on to the next level. You would probably create a problem that’s more challenging for them. If that child was having difficulty, then you probably would not give them the solution directly. What you would do is you’d give them a hint, maybe a little help, and you would hope that from that hint or from that little bit of help, they would be able to come to the actual solution. And if they weren’t able to come to the solution, to keep them from being frustrated and to keep you from being frustrated, you’d probably give them the answer, but you would make a mental note of that. You’d remember that you had to give them the answer to that problem, and so the next time you came to that student, you would ask them a problem that was similar to [the previous one] and hope that they could remember what you told them, and that would overcome that area that they are having difficulty in. And that’s exactly what the computer lab does for them. It monitors every keystroke. It monitors how much time they take to solve a problem. It monitors the mistakes they are making and it monitors how many times they have to ask for help. And then it generates problems based on that. And during the computer lab time, the students that are having the most difficulty are the students that I spend the most of my time with. And those students that are not having as much difficulty become my aides.

Transcript from Tremain Nelson

Those students that perform well in the lab are not always the same students that performed well in the classroom. And the reason is personality, as well as learning styles. Some students prefer working at their own pace in the computer lab, where they are able to go and do the work at their own pace. For those students that don’t feel comfortable with a pencil or paper and a desk, it’s a break for them, because they have an opportunity to do something that is interactive. It’s fun and exciting for them because they are able to actually participate in the learning a little more directly. The biggest change over the year that I’ve seen with these students since they’ve been working in the lab is their enthusiasm, their motivation mainly about mathematics itself. The lab is open in the morning and it is not mandatory for these children to come up to the lab, but often they do – so much so that we’ve had to bring the teachers on to work the lab during the morning time so that we can provide someone there for those students that want to come in the morning. And we have a full lab just about every morning.

One of the biggest benefits of using self-paced computer technology is that students at various levels receive the instruction that they currently need. In a regular classroom, the teacher decides the content that will be taught each day, and all students are expected to participate in the lesson and eventually master the content, even if their prior knowledge is inadequate. That can pose real challenges. But computer-based software that monitors student achievement allows students to progress when they are ready. As Tremain says, “You no longer have to teach to the middle ground.”

Transcript from Tremain Nelson

When you think about a traditional classroom, when you ask a question and you are direct teaching from the front of the class, you want someone to say the right answer. And as soon as that child says the right answer, you assume that everybody else in the room knows that answer and agrees with that student, and then you move on. And what you end up missing is that child who was in the back of the room, who didn’t know that that was the right answer, and was too afraid to speak up and say, “I don’t understand.” Well, in the computer lab, since it’s self-paced, then I know and that student knows, in real time, whether or not they understand whatever concept we’re currently discussing. What ends up happening is I’m able to now go over to that child, or to send a student who is performing well over to that child, and spend some quality time with him trying to get him to the level where they are actually able to produce and do well.

Reflection:

How might computer-based assessment, as well as other forms of individualized assessment, improve student achievement in your classroom?

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Insights Into Algebra 1: Teaching for Learning

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