Formative and Summative Assessment of Objectives

In looking at the California Common Core Standards for Algebra 1 I realized that the standards are written in a way that is not that compatible with performance-based assessments. The standards are broken down in a compartmentalized fashion inconsistent with how one might use the skills described in real-world situations. When designing or analyzing things, one checks the work by switching between the text representation (specification), the mathematical model, the computer simulation, and graphical output to ensure things are consistent. In the event a mistake/error is suspected, one must troubleshoot, which could involve checking the Algebra by hand. One works between disciplines and between representations of the Mathematics based on judgement and experience. Good Performance-based assessment should mirror that process or aspects of it. The standards at this level are more basic because you need the student to build a certain level of fluency and automaticity in their work. You have to teach the student how to use each individual tool well before you teach them how to choose the right tool for the design/troubleshooting process. Algebra 1 is an introductory course with an emphasis on teaching the use of the individual tools, not the selection of the tools. The difference in scope between the Standards and the requirements of this assignment means that I will present an Performance-based assessment that actually spans several standards in Algebra 1, but is a more authentic situation for the student. One consequence of the introductory nature of Algebra 1 is that many of the standards are not geared towards higher levels of Bloom's Revised Taxonomy as many of the standards deal with procedural skills. Consistent Meta-level learning tends to come after the student reaches Geometry, Trigonometry, and Calculus.  I tried to choose the standard so that there is a chance at higher levels in the Taxonomy, but I also had to choose something consistent  with the ELL learning needs of my students. Right now I have students in Grade 8 coming from Chinese Public schools who can barely ask to go to the bathroom in a complete sentence. Choosing a performance-based assessment that they can comprehend is difficult or impractical as the student may not be able to comprehend most of the assignment. In addition, simplifying the language of a performance-based assessment will remove some of the complexity that makes the task authentic and "real world". So I am consciously not using a performance-based assessment at this point because I feel postponing it until we cover more standards in the course and have more of the students at a point where their language acquisition is at a level where they can operate with the independence needed to benefit from a performance-based assessment. I will present a summative assessment sequence that is realistic for most of the students I have, then present the performance-based assessment that spans several standards for the sake of meeting the requirements of this assignment.

For the eighth graders I teach, all but two have very strong mechanical skills as shown by their diagnostic results from the start of the year. Consequently, I do not need spend a lot of time explaining how to solve equations. In addition, the mechanics of solving an equation is not a high level task in Bloom's Revised Taxonomy. While this post may present things in a way that suggests I do not teach how to solve an equation, the mechanical skills are revised in parallel with the work described here. I am not including how I teach/revise the mechanics of the algebra for the sake of brevity.

The Standard:

"A1 A-CED: Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA*" (California Department of Education, 2013)
CA means an extra standard added by California. *Means the standard includes working from context (Real-World problems).
In order to limit this post to something that would be the subject of a single unit, I have to narrow the standard by not including quadratic, exponential, and rational relations as these require considerable time to teach and would result in a unit that is long enough to basically be a Semester. So I will narrow it to: Create equations and inequalities in one variable including ones with absolute value and use them to solve problems with context.

Now one needs to be aware that in the context of creating mathematical models to represent real-world situations, the word "create" means the analysis of text and translation of text to equivalent Algebraic equations and the application of Algebraic rules to obtain a correct solution. Thus the teacher has to teach the mechanics of performing the algebra and the skill of analyzing and creating the model from text. It is not the same meaning as understood in terms of Bloom's Taxonomy.

I will describe the summative and formative assessments I use in teaching the skills needed to analyze and create the algebraic model from text.

Formative Assessment: 

Formative assessment will be done in three tracks, scaffolded worksheets, observing students working and spot checking their work, and using word jumbles for struggling students.

There is a whole series of sheets for formative assessment starting from ones that deal with phrases and expressions, before progressing to entire sentences. Examples of the sheets are shown below, you can contact me for full sheets if you want them. All students answer these sheets.

Figure 1: The Basic Phrases Worksheet

Figure 2: Basic Phrases 2 Worksheet.

Figure 3: Basic Expressions Worksheet

Figure 4: Related Variables Worksheet

Figure 5: Sentences Worksheet

Along the way I will question students about certain aspects of their work relating to the following:

• Understanding words that relate to brackets tend to connect two quantities. The sum of a number and 7 is increased by a factor of ten = (j+7)x10 rather than j+7x10. because a sum contains two quantities.

• Confusion of "twice a  number" and "a number squared" 
• Confusing words for place value like hundreds and hundredths.
• Getting the order right for "subtracted from" statements.
• The term "average of" is a combination of operations.
• Inserting the words "by a factor of" change addition and subtraction to multiplication and division.
• More technical words like subsequent, preceding, prior, increment, and decrement.

While they work I will spot check students by taking the answer (equation) written by the student and working backwards with the student to see if we can get the same English that he/she translated from. Students will be encouraged to use this technique to check their own work or the work of peers.

If I feel a student needs more work, I will then give them a word jumble. Giving them a handful of slips of paper with the English phrases that relate to the basic operations (addition, subtraction, multiplication, and division) and another handful of slips with the Algebraic phrases and having them match them up will help check if they are able to translate correctly. One that I use is shown below. A struggling student is told to stop work on a worksheet and work on this instead. 

Assessment of this is straightforward as it is usually quite apparent what is correct, and that most students will try and get to a point where they make no mistakes. There will be several versions of the word jumble starting with basic terms, then one starting with a mix of basic and academic vocabulary, and one of exclusively academic vocabulary. The one I use will depend on the whether a student struggles with phrases, expressions, or sentences. There are a limited number made, so that kids are not copying the work of others. Matching games like this bias kids towards 100% completion, as few kids in eighth grade want to have things that don't match. 

In addition a reflection assignment is given in the middle of the summative assignment sequence. The reflective assignment is described below as it fits within the summative assessments.

Summative Assessment

The summative assessment is made from two parts: a traditional test and a worksheet of more complex situations. The questions of the assessment that look at the translation skills are shown below:

Figure 6: Summative Test Excerpt

After this assessment is graded, it is returned and they reflect on it using guidelines for reflecting on how they prepare for an assessment and how they worked during the assessment. This division is based on the work of Deanna Kuhn and her work on Meta-strategic and  Meta-task knowledge (Kuhn, 2000). Excerpts are shown below of the reflective guidelines. The reflection is considered a Formative assessment.

Figure 7: Reflective Guidelines

They are then given a couple of days to prepare for the more complex problems which are meant to increase exposure to more real world problems and expose them to situations foreign to their experiences in China. There are two versions, one for this point in the year, and one later in the year after we have done a unit on graphing to link the two units. Students are given two periods to do it. All students answer the assignment. The assessment for this point of the year is shown below:

Figure 8: Complex Summative Task (Carnegie Learning, 2011)

Performance-based Assessment:

This assessment combines graphical representations, creating models from text and drawings, 

checking the validity of the model, obtaining a final answer, and checking it's validity. Note that the standard only asks for part of what is described in the previous sentence. As stated in the beginning of this post, you generally won't get Performance-based assessment for the narrow focus of the standard. You also have to build the language skill of the students to a point where they can work on their own.  This assessment will be given to all students.

Figure 9: Performance- based Task.

Works Cited

California Department of Education. (2013, August). California Common Core State Standards Mathematics Electronic Edition. Sacramento: California Department of Education.

Carnegie Learning. (2011). Lesson 1.1. In Math Series Course 3 Assignments (pp. 1-14). Carnegie Learning.

Kuhn, D. (2000). Metacognitive Development. Current Directions in Psychological Science, 9(5), 178-181.