Effective technologies for learning and doing mathematics should be based on sound cognitive theory, be empirically tested against alternatives, and be primarily addressed at mathematics as a modeling language.
Before turning to a summary of Cognitive Tutors in general and PAT in particular, I want to make a few general comments about the role of cognitive theory and empirical testing in the development of technology-enhanced learning innovations and the role of mathematical modeling as the appropriate core focus of innovations for mathematics.
Why is it important that we perform empirical tests of our innovations in comparison with alternatives? Our intuitions about learning are biased by limited information -- overly influenced by our memories of our conscious learning experiences. Not every experiment can be run comparing alternative features of instruction and their interactions. Although technology has mastered calculation of various kinds, arithmetic, graphic, symbolic, logical, humans are the only masters of translating problems into mathematics, building theories, producing communicative forms. In applying the ACT theory to instruction, we have focused on the idea that human one-to-one assistance or tutoring is extremely effective in facilitating learning. The cognitive model is also used to trace students knowledge growth across problem solving activities.
Cognitive Tutors have been subject to comparative evaluations in the lab and in classroom for more than 12 years (Anderson, Corbett, Koedinger, & Pelletier, 1995).
Two experiences led us to take a different "client-centered" approach when we began to develop the Pump Algebra Tutor. Table 1 shows part of day one activities of a two day performance assessment used in the Pump Algebra curriculum.
To reach the goal of creating improved instructional supports to help students learn to be successful on assessments like the Photographer Career problem in Table 1, we began to research issues of mathematical modeling and the underlying competencies required. This hypothesis followed from my prior cognitive science research on the importance of inductive experiences in the evolution of geometry knowledge (Koedinger & Anderson, 1990) and from observations of students who can successfully solve concrete result-unknown questions, like 2 and 3, but cannot produce the corresponding algebraic sentence. This production rule characterizes tacit performance knowledge for composing algebraic "embedded clauses" for a quantity, like 42h + 35 for the total bill (Q1), from knowledge of simple clauses, like 42h for the hourly charge (Q2) and 35 for the service charge (Q3). The rationale for having students do result-unknown questions before symbolization is this: By stepping through the arithmetic operations with concrete instances of quantities, students have a source from which to learn, by analogical induction (Anderson, 1993), production rules for producing algebraic sentences using the same operations and quantities.
We used an early version of PAT to implement an experimental learning study in which we compared a control condition in which students solved problems in the Forester textbook format to an experimental group in which students solved same problems but with the questions rearranged so that the concrete result-unknowns (#2 and #3) would first, before the symbolization question (#1). The general point here, consistent with Paul Goldenberg's paper, is that students need to not only learn mathematical concepts, but also develop a mastery or fluency with mathematical modeling languages like algebra, programming, statistics notation, dynamic geometry tools, etc. As part of the development of the Pump Algebra Tutor (PAT), Pittsburgh teachers wrote problem situations, like the Photography problem above, intended to be personally or culturally relevant to students.
Students begin work on PAT problem situations by reading a description of the situation and a number of questions about it. This provision of timely feedback is a critical feature of Cognitive Tutors that leads to substantial cognitive and motivational benefits. In addition to just-in-time feedback, a second way PAT individualizes instruction is by providing context-sensitive hints. The Equation Solver window (lower-center) shows how the student solved a similar question (question 3). By keeping students engaged in successful problem solving, PAT's feedback and hint messages reduce student frustration and provide for a valuable sense of accomplishment.
The activities in PAT are organized hierarchically so that related problem situations that draw on a core set of skills are organized in to "sections" and then sections that use the same set of notations, tools, and broader concepts are organized into "lessons". In the classroom, students work together in groups or teams to solve problems similar to those presented by the tutor. The Pump curriculum uses alternate forms of assessment including performance tasks, long term projects, student portfolios, and journal writing.
Following the observations of Schofield, Evans-Rhodes, & Huber (1990) and Wertheimer (1990), we have also observed the impact of the use of PAT on changes in classroom social and motivational processes.
In the computer lab, teachers are glad to essentially have a teacher's aid for every student and thus are freed to be facilitators and provide more one-on-one instruction with individual students. How and why does Cognitive Tutor use facilitate the spread of effective teaching principles and practices and the institution of curriculum reform?
A DoE-FIPSE project in which we adapted PAT for use in college-level developmental math courses has provided the opportunity to observe PAT-inspired changes in curricula and teacher practices over multiple semesters. I believe that effective technologies for learning and doing mathematics should be based on sound cognitive theory, be empirically tested against alternatives, and be primarily addressed at mathematics as a modeling language. Cognitive Tutors, like PAT, have the potential not only to dramatically increase student achievement, but also serve a professional development function for teachers. September 4, 2013 By Meeghan Mousaw Leave a Comment Pin +1 Tweet Share Stumble“Puppy, so here you are with the six pigs! These Top Free Reading Apps are a fun way for K-5 students to practice their reading skills! These definitely can come in handy during reading workshop, reading time at home, tutoring sessions, long road trips, etc.
4. Grammar Wonderland (Elementary) - Practice using nouns, adjectives, verbs, and more in this fun app! I will discuss the status and promise of research-based "intelligent" systems that support students in mathematical modeling and tool use. What is the technological infrastructure that one can assume most schools and teachers will have access to in years 2000-2005? In particular, I believe that effective technologies for learning and doing mathematics should be based on sound cognitive theory, be empirically tested against alternatives, and be primarily addressed at mathematics as a modeling language.
If the intuitions and beliefs that guide our design of learning environments were fully informed and perfect, there would be no need for such experiments. We are subject to what I call "Expert Blindspot" -- as experts in a domain we are often poor judges of what is difficult and challenging for learners. Learning how to create mathematical models of problem situations is difficult but it is the key to mathematical success in our modern world. Bloom (1984) showed that an individual human tutor can improve student learning by two standard deviations over classroom instruction. The knowledge tracing technique is dynamically updating estimates of how well the student knows each production rule.
A cognitive tutor for writing programs in the LISP computer language was compared to a control condition in which students solved the same programming problems without the aid of the cognitive tutor. First, at the time of the geometry studies, the NCTM Standards were coming out and suggesting a deemphasis on proof in high school geometry.
The goal of day one is not for students to produce a particular numerical answer, but rather to produce an analysis and a representation of that analysis in mathematical forms including tables, graphs, equations, and words. Completing high school and attending a four year college program in photography is the third option. Since you are an excellent algebra student you want to use algebra to make this analysis, and then use this analysis to write a letter to your friend explaining clearly the advantages and disadvantages of each option. In particular, we focused on studying what students know and do not know about symbolic modeling. Hadley had been using a similar problem format, but put a particular emphasis on problem contexts that would be more authentic to students and that contained real data.
Such students already have effective performance knowledge or production rules for comprehending the English problem statement, for extracting the relevant quantities and quantitative relations, and for a producing a numerical answer. I call this use of concrete instances to help students induce algebraic sentences the inductive support strategy. They investigate the situation by representing it in tables, graphs, and symbols and using these representations to answer the questions. This problem appears in later stages of the curriculum after students have acquired some expertise with constructing and using graphs and tables for single linear equations. During that time, the tutor monitors their activities, and provides feedback on what they are doing. In a parametric study with the LISP tutor, Corbett and Anderson (1991) provided a demonstration of how the timely feedback leads to dramatic reductions in the learning time needed to reach the same level of post-training performance. Using the model tracing mechanism described above, PAT is always following each student's particular approach to a problem. The current focus of attention is based on the selection of the worksheet cell for question 4, under the column entry for 'miles driven' -- this cell is highlighted in Figure 2. If needed, a further message provides the algebraic equation that the student needs to set up and solve.


The student enters their own equation and solves it by indicating standard algebraic manipulations.
In addition to these functions of model tracing, PAT provides learning support through knowledge tracing.
In the current PAT curriculum (used in schools in the 1997-98 school year) there were 22 lessons each of which contained about 4 sections on average and each section contained about 5 required problems and about 5 additional problems. The typical procedure is to spend 2 days a week in the computer lab using PAT and 3 days a week in the regular classroom.
Teams construct their solutions by making tables, expressions, equations, and graphs which they then use to answer questions and make interpretations and predictions. This experience is eye-opening for many teachers who may see new aspects of student thinking and feel the advantages of greater student-centered learn by doing.
Is there something special about Cognitive Tutors that makes such spread more likely than alternative educational technologies like books, traditional CAI, or malleable tools? These changes were accompanied by significant quantitative improvements in student learning. I have argued for and illustrated these points in the context of Cognitive Tutors and in particular, the Pump Algebra Tutor (PAT). Because of the cognitive model, Cognitive Tutors can provide a living example of effective instructional practices.
Aesop for Children – This app provides over 140 classic fables accompanied by illustrations and interactive animations. Phonics Genius – This app is loaded with over 6,000 professionally recorded words that are grouped into 225 phonics categories. There are catchy animated songs and rhymes to make learning about nouns, verbs, sentences, etc.
Such systems are beginning to reshape the mathematics classroom, the way teachers teach, and what and how students learn. I illustrate these points in the context of an educational technology we call "Cognitive Tutors" (Anderson, Corbett, Koedinger, Pelletier, 1995).
Unfortunately, our intuitions and beliefs about learning and instruction are limited and are not always accurate.
Perhaps few would disagree about the importance of evaluating our educational innovations to better understand how they do or do not improve on current practice. Such a guide should help us prune design ideas not likely to enhance learning and inspire new ideas that will. In other words, the average tutored student is performs better than 98% of students receiving classroom instruction. These estimates are used to select problem solving activities and to adjust pacing to adapt to individual student needs. Second, we had experienced the importance and difficulty of integrating the technology with the classroom and paper-based curriculum.
In other words, the students goal is to create mathematical models of the alternative situations presented in the problem.
This technical program costs about $18,000 and the average salary for those completing the course is $1925 per month. The average cost for a four year program is $50,000 and a graduate can expect to earn about $2745 per month.
However, production rules for "speaking algebra", that is, for taking a problem understanding and expressing it in algebraic symbols, are either weak or missing (cf.
Students in the Inductive Support experimental group learned significantly more from pre-test to post-test than students in the Textbook control group (Koedinger & Anderson, in press).
These problems were added to PAT using an intelligent problem authoring system in which teachers type the problem description and enter an example solution (Ritter, Anderson, Cytrynowitz, & Medvedeva, 1998). Helping students to understand and use multiple representations is a major focus of the tutor. In addition to cognitive benefits, there are also motivational benefits of timely feedback. At any point in constructing a solution, a student can request a hint and PAT will provide one that is sensitive to what the student has done up to that point. Initially, students explore common situations involving positive quantities, mostly whole numbers and some simple fractions and decimals, and represented these situations mathematically in tables and graphs. In the classroom, learning is active, student-centered, and focused primarily on learning by doing. Teachers play a key role in helping students to make connections between the computer tools and paper and pencil techniques and to see how the general concepts and skills for representation construction and interpretation in each case are the same. PAT may enhance student motivation for different reasons: 1) authentic problem situations make mathematics more interesting, sensible, or relevant, 2) students on the average would rather be doing than listening and the incremental achievement and feedback within problems has a video-game-like appeal, 3) the safety net provided by the tutor reduces potential for frustration and provides assistance on errors without social stigma, 4) the longer term achievement of mastering the mathematics is empowering. Unlike other educational technologies, Cognitive Tutors have a running model of student thinking and of adaptive student-centered instruction.
PAT was initially used at the University of Pittsburgh in the Fall semester of 1995 as an add-on to a traditional "Intermediate Algebra" course.
PAT is based on the ACT theory of cognition and a production rule model of student problem solving and mathematical modeling. Teachers working in the computer lab have more time to observe student performance on thought revealing problems and have observe learn-by-doing instruction in action. The student listens to the word being read, and then has to touch the word that matches from the choices given.
Also, check out Grammar Wonderland (Primary) for the same kinds of fun games for younger learners. Cognitive Tutors are based in computer science research on artificial intelligence techniques and cognitive psychology research on the nature of human learning and performance. One problem is that our intuitions are based largely on our conscious learning experiences, but a great fraction, perhaps the majority, of what we learn is at a level below our awareness.
Nevertheless, I think it is worth emphasizing the danger of being biased by expert blindspot and lured by our personal intuitions into assuming that our educational innovations and reforms will necessarily be for the better. Cognitive theory also provides a way to accumulate reasons for past successes and failures to inform future practices. This results provides a sort of "gold standard" for comparing the effectiveness of educational technologies. The LISP tutor allowed students to engage in productive problem solving search, but reduced unproductive floundering. Thus, in the Pump Algebra Tutor project we designed the tutor and the curriculum hand-in-hand. These models are then used on day two to answer specific questions about these choices and to write a full page recommendation using the models as support for the recommendations made.
We began to experiment with different approaches that might better aid students in learning to model with algebraic symbols. As I began to observe and analyze student thinking toward creating a cognitive model, I formed the simple instructional hypothesis that having students answer the concrete "result-unknown" questions 2 and 3 might facilitate their learning to translate such problems into the language of algebra (question 1). The authoring system has an incomplete and imperfect model for reading English text, but can make reasonably guesses as to how quantities in the entered solution map to phrases in the text. The problem involves two rental companies, Hertz and Avis, that charge different rates for renting large trucks. Like good human tutors, PAT's tutorial interactions with students are brief and focused on individual students' particular learning needs as they come up in the context of problem solving.
Much like the motivational attraction of video games, students know right away that they are making progress and having success at a challenging task.
The tutor chooses a hint message by using the production system to identify the set of possible next strategic decisions and ultimate external actions.
The difference is that, in PAT, these examples come in the process of problem solving when the student is better able to understand and make use of the example. As the year progresses more complex situations are analyzed involving negative quantities and four quadrant graphing is introduced. Teachers spend less time in whole-group lecture and more time in individual and cooperative problem solving and learning.
Literacy is stressed by requiring students to answer all questions in complete sentences, to write reports and to give presentations of their findings to their peers.


At the end of each semester these tasks are graded by the teachers at a mini-scoring conference where teachers come together, construct a scoring rubric, and score all the student papers in an afternoon. We have demonstrated this impact in experimental field studies in city schools in Pittsburgh and Milwaukee, replicated over 3 years, and targeting both 1) higher order conceptual achievement as measured by performance assessments of problem solving and representation use and 2) basic skills achievement as measured by standardized test items, for instances, from the math SAT. Thus, the system provides an active "living example" of research-based principles and practices. As shown in the left of Figure 3, this supplementary use led to significant improvement in students' problem solving abilities over a traditional course without PAT (45% vs. In this way, Cognitive Tutors can carry research-based practices into the classroom and serve as change agents for professional development. Word lists are provided, but you can also add your own words and even record your voice reading them! Cognitive Tutors have been created to help students learn in a variety of mathematics and computer programming domains and have been subject to laboratory and classroom evaluations that demonstrate the potential for dramatic learning gains from appropriate use of this technology. The grammar rules of our first natural language, English in my case, are an excellent example. Two different Cognitive Tutors for geometry proof design were used in classroom studies compared to control classes using a traditional geometry curriculum without the cognitive tutor.
A high school math teacher, Bill Hadley, and a curriculum supervisor, Diane Briars, had been working on an algebra curriculum to make algebra accessible to more students, to help students make connections between algebra and the world outside of school, and to prepare students for the "world of work" as well as further academic study. As Paul Goldenberg has said models provide a "general answer" to problems that can be used to answer multiple specific questions of the kind usually asked in math class. Students investigate the problem situation using multiple representations and computer-based tools, including a spreadsheet, grapher, and symbolic calculator -- in Figure 2 these are the Worksheet, Grapher and Equation Solver windows, respectively. When a student is having trouble, PAT does not immediately provide correct answers or even detailed advice.
It chooses among these based on the student's current focus of activity, what tool and interface objects the student has selected, the overall status of the student's solution, and internal knowledge of relative utility of alternative strategies. For example, a student who was skilled in writing equations with positive slopes and intercepts, but had difficulty with negative slope equations would be assigned problems involving negative slopes. Similarly, as situations increase in complexity, formal equation solving and graphing techniques are introduced to enable students to find solutions. The teacher plays a facilitator role, but may do some whole-group lecture and discussion to highlight student discoveries or respond to recognized needs of the class as a whole. Because each teacher scores papers from every other teachers' class as well as their own they come to have a better understanding of the objectives of the curriculum, what students know and do not know, and in what ways other teachers' students may differ.
In comparison with traditional algebra classes at the same and similar schools, we have found that students using PAT and the Pump curriculum perform 15-25% better than control classes on standardized test items and 50-100% better on problem solving & representation use. Furthermore, by essentially serving as an extra teacher's aid for each student in the classroom, Cognitive Tutors free teachers to observe individual student thinking more often and more closely and to reflect on their instructional practices in the context of more one-to-one interaction with students. 30% as measured by a performance-based Cell Phone assessment much like the Photographer assessment in Table 1).
In a formative laboratory study with PAT we showed that we could improve students learning of symbolic modeling skills over a textbook approach by having students first solve concrete arithmetic versions of problems before abstracting these operators in an algebraic model.
A developmental model for algebra symbolization: The results of a difficulty factors assessment. Illustrating principled design: The early evolution of a cognitive tutor for algebra symbolization. Artificial intelligence in the classroom: The impact of a computer-based tutor on teachers and students. We learn these rules, in the sense that they determine our behavior in language comprehension and production, well before we are consciously aware of them.
In both studies, students in the experimental classes scored 1 sd better than students in control classes. We teamed up with Hadley and began evolving the curriculum and designing the tutor by sharing ideas from both research and practice.
Students construct the Worksheet (lower-left of Figure 2) by identifying the relevant quantities in the situation, labeling the columns, entering the appropriate units, entering algebraic expressions, and by answering the numbered questions in the problem description.
Instead, PAT tries to maximize student opportunities to discover or reinforce appropriate concepts or skills on their own.
Successive levels of assistance are provided so as to maximize the students' opportunities to construct or generate knowledge on their.
Knowledge tracing can also be used for "self-pacing", that is, the promotion of students through sections of the curriculum based on their mastery of the skills in that section. Systems of linear situations and quadratics are developed through the introduction of situations in which they naturally occur, for example, modeling and comparing the price structures of two rival companies that make custom T-shirts. Student responses in such close interactions provide teachers with immediate and detailed feedback on the effectiveness of their practices that they can then use to adjust those practices. In the Spring of 1996, instructors used the PAT authoring tool to create new problems better adapted to the particular needs and interests of their students.
In field studies of the use of PAT and the Pump curriculum we demonstrated the combination leads to dramatic increases in student learning on both standardized test items (15-25% better than control classes) and new standards-oriented assessments of problem solving and representation use (50-100% better than control classes). PAT is in regular use now in over one hundred classrooms at the middle school, high school, and college levels. Students construct the graph of the problem situation (upper-right) by labeling axes, setting appropriate bounds and scale, graphing the lines, and identifying the point of intersection.
PAT provides two kinds of assistance: just-in-time feedback on problem solving steps taken and on-demand hints on next steps needed.
Modeling vertical motion and area situations provide contexts for introducing and using quadratic functions. The approach of PAT and the Pump curriculum focus is on developing students competence in creating mathematical models of problem situations rather than on answers to isolated questions.
Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.
Many of the 40 schools using PAT are in big cities, involve average teachers, and include a large of number of disadvantaged minority or learning disabled students.
The Equation Solver (lower-center) can be used at any time to help fill in the spreadsheet and identify points of intersection. Instructors began to use PAT problems in their regular classes and began to experiment with more student-centered learn by doing outside of the computer lab.
Mathematical model provide the power to garner a deeper understanding of a problem situation such that multiple, unanticipated questions can be addressed and answered. A second lesson came from a third party evaluator who studied changes in student motivation and classroom social processes as a consequence of the use of the Geometry Proof Tutor. The student can use these representations to reason about real-world concerns, such as deciding when it becomes better to rent from one company rather than another. The consequence of these new practices was an increase in end-of-course achievement beyond that found in the experimental classes in the Fall (65% vs.
In fact, there is ample evidence from cognitive psychology research that our brains continue to engage in implicit learning processes. Schofield, Evans-Rhodes, and Huber (1990) found the classroom evolved to be student centered with the teacher taking a greater facilitator role supporting students as-needed on the particular learning challenges each was experiencing.
This point was repeated in a Math Teacher article by one of the participating teachers in which Wertheimer (1990) emphasized that because the cognitive tutor was effectively engaging students he was more free to provide individualized assistance to students who needed it. Over the next two semesters, these practices evolved and PAT use became better integrated with further modest increases in student learning over past semesters (68%, 71%). In preparation for the Spring of 1997, the University of Pittsburgh made a decision to fully reform their intermediate algebra course to more generally target "quantitative literacy". Lecturing was deemphasized in favor of more "workshop" time in which students worked on projects in collaborative groups. The typical college level approach of a lot of instructor-centered lecture time and a little student-centered recitation time was replaced with a lot of student-centered workshop and a little whole-group reflection and targeted lectures. The consequence of these changes, in the first semester of their implementation, we yet another increase in end-of-course student performance (79%).



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