Introduction: The algebra framework

The algebra framework is a tool designed to help teachers gather, consider, and act upon specific information about how their students understand algebra.  The framework is made up of an interconnected series of components that includes an algebra learning model, diagnostic tests, an interpretation guide with scoring rubrics, and practical instructions.

The algebra framework was created because teachers wanted more information about how their students were constructing understandings of the important ideas algebra.  We designed the framework with the assumption that in high school algebra, letter symbols are used in three ways: as generalised numbers, unknowns, and variables.  At this time, the algebra framework does not address the use of letter symbols as parameters.

Components of the algebra framework

The algebra learning model

The algebra learning model was created by the research team to describe the different ways students demonstrate important algebraic ideas before they have a complete understanding of algebra.  The algebra learning model represents the process of how many high school students develop understandings of algebra.  The algebra learning model was developed from findings of previous research conducted by members of our team and from international research literature.  We found that as students learn ideas about algebra, they demonstrate a range of problem solving strategies.  Because students demonstrate these strategies in predictable ways, the strategies students use could be classified into stages of learning.

The algebra learning model was used to design diagnostic tests, scoring rubrics an interpretation guide and practical instructions for teachers to use in their mathematics classrooms.  The algebra learning model contains ideas that we are continually reconsidering as we use them in practical ways with our students.  We welcome your feedback as you use the components of the algebra framework.

Algebra is too often regarded, both by students and teachers, as a new topic concerned with letters rather than numbers that has no connections to previously learned mathematics.  However, the NZ Curriculum now has combined Number and Algebra as one strand and there is a wealth of international research that has documented intimate connections between algebra and other branches of mathematics, particularly arithmetic.  Some of our previous work has shown that knowledge of arithmetic structure, knowledge of inverse operations and understanding of equivalence are important prerequisites for learning algebraic strategies.  Knowledge of basic facts is absolutely crucial!  In the past we have regarded this knowledge as arithmetic, but we are now beginning to viewing it as early algebra.  The Numeracy Development Projects (NDPs) have been a strong influence on our efforts because they give us a clear perspective of the strategies that children use in arithmetic and the knowledge required for these strategies.  We have been examining the strategies that students use to solve equations, find relationships and express generality, and what prerequisite knowledge they use when doing so, and have found clear stages of learning.

Diagnostic Tests

Four diagnostic tests were developed from the algebra learning model by the research team to help teachers have practical ways of gathering information about how students understanding algebra.  Specifically, teachers wanted more specific information about the strategies students use with letter symbols when they represent generalised numbers, unknowns, and variables.  The diagnostic tests were developed to inform teachers about what strategies students can or cannot use with letter symbols rather than to generate an overall test score.  These four diagnostic tests have been used by teachers on our research team in high school classrooms with their students.

The first diagnostic test examines how students understand unknowns in the context of solving equations.  The second diagnostic test examines the prerequisite knowledge students are required to master before they can learn many of the algebraic letter symbol strategies of algebra.  For students to use these strategies successfully, we have found that basic facts knowledge is absolutely crucial.  Students need to be at stage 6 or above on the New Zealand Numeracy Development Project Number Framework for basic facts.  The third diagnostic test examines how students can use letter symbols to express generality.  The fourth diagnostic test examines if students can use multiple representations to describe relationships and whether they can find a relationship between two variables or just use the iterative pattern within one variable.

The diagnostic tests can be administered as “One-on-one” Diagnostic Interviews or as Whole Class Written Diagnostic Tests.  Instructions for interviewing individual students or testing whole classes have been written in a step-by-step way to help teachers who are new to using the diagnostic tests. 

We suggest teachers follow the instructions carefully so that the results can be discussed meaningfully with other teachers who are also using the algebra framework.  By engaging in discussions with other teachers, we noticed that we were better able to select appropriate ways to help our students learn the challenging ideas about algebra.

• Link to the diagnostic tests.

Interpretation Guide with Scoring Rubrics

An interpretation guide with scoring rubrics was developed by the research team to help teachers make specific judgments about what strategies and knowledge students are demonstrating in the four diagnostic tests.  Teachers can then used these judgments to make changes to their planning and interactions with students. The interpretation guide uses ideas from the algebra learning model to help teachers make specific judgments about what strategies and knowledge students need to learn next and what they should emphasise in their planning.

Scoring rubrics help teachers classify students’ strategies used for solving equations and more specific knowledge about generalised numbers, relationships and prerequisite knowledge.  The scoring rubrics have been written in an easy-to-use way. The scoring rubrics look very similar to the diagnostic test and follow the test questions in order.  The scoring rubrics contain information about the types of responses to expect from students and how to score those responses.  To interpret the scores, teachers will be required to consider several test questions at the same time.  Combining the scores of several questions will show teachers the level of mastery students demonstrate about a particular problem solving strategy or knowledge topic.

Templates have been designed and provided for teachers to record, summarise, and interpret the class set of information generated by the diagnostic tests.

• Link to the Templates.

Practical Instructions

Practical Instructions were developed from the experiences of the research team using the components of the algebra framework in high school classrooms with students.  The practical instructions are written in a step-by-step way to help teachers who are new to using the algebra framework.

We suggest teachers follow the practical instructions carefully so that the test results can be discussed meaningfully with other teachers who are also using the algebra framework.  By engaging in discussions with other teachers, we noticed that we were better able to select appropriate ways to help our students learn the challenging ideas about algebra.

Getting Started

A great place to start for teacher new to the algebra framework is with a whole class written test using the Solving Equations Diagnostic Test.  This starting combination is powerful because it gives teachers a quick overview of how their students understand unknowns in the context of solving equations. The instructions for administering the Solving Equations Diagnostic Test are included with the printer-ready versions of the diagnostic tests listed in the Diagnostic Test section of this website.

After teachers have scored, and interpreted set of written responses from their students, they will a good sense of how their students understand the ideas presented in the diagnostic tests.  Teachers will be able to use this information to inform their lesson planning.  Teachers may wish to have more specific information about why a particular response was offered by a number of students or why a particular student responded they way he or she did.  At this point, we encourage teachers to conduct One-on-one Diagnostic Interviews with students, which are described in the Going Deeper section below.

Going Deeper

For teachers interested in understanding more about how students are thinking about algebra, we recommend teachers conduct a One-on-one Diagnostic Interview with a student.  The interview gives teachers the experience of hearing and seeing the levels of mastery of basic facts and algebraic notation directly from their students.  Teachers should repeat the process with up to six students so they can get a sense of the range of understanding demonstrated by their students.  Please note that the One-on-one Diagnostic Interview uses questions that are similar but not identical to the Solving Equations Diagnostic Test.  The test questions are different so that you can administer both tests to the same students so they will not be repeating the test.

• Link to the page for instructions to administer the One-on-one Diagnostic Interview.

Even though it may seem like an unnecessary step, teachers on our research team found that conducting several one-on-one diagnostic interviews with students was an excellent investment of their time.  Teachers also noticed that the diagnostic interviews revealed information about student understanding of algebra that was not evident to them on the written test alone.  The diagnostic interview uses prompts such as “How did you work that out?”, “Explain your thinking.” and “How do you know that?”  Such questions allow teachers to focus on the experience of one student engaging with a diagnostic test at a time.  This ‘one-on-one’ experience allows teachers to hear and see what the strategies and knowledge topics students find challenging to learn in language that the student uses.

After conducting about six One-on-one Diagnostic Interviews, teachers will have a good understanding of how of the components of the algebra framework work together.  We recommend that teachers use the other diagnostic tests with their students and use the results to inform their lesson planning.  The other three diagnostic tests are shorter that the Solving Equations Diagnostic Test and the administration instructions are included with the printer-ready versions of the diagnostic tests listed in the Diagnostic Test section of this website.