What does the student response data of the following question in Ei ASSET[1] say? What are the errors committed by students?

2u – 9 = 12 – u.

What is u?

Students make different types of errors in carrying out a procedure other than computational errors while solving some of the Mathematics problems as evident from the above case and student response data of many such Ei-ASSET questions. What should be done to enable grade-appropriate learning and reduce the extent of students not making such errors? Learning instructions which lead to reduction in the extent of such errors are desired. However, it would not suffice as some learners will still make errors. Researchers like Block and Burns[1] have shown that when students’ learning difficulties are identified, corrected and reinforced, there is an increase in their cognitive gains. Components of mastery learning such as feedback and the process of leading learners to be aware of their errors and engaging in possible correction has led to significant results as reported in many past research studies, one of them by Burrows & Okey[2]. It helps if a teacher/expert instantly shares feedback on these errors and helps each such learner correct his/her errors. However, how practical is that in a classroom scenario to comprehend student work, identify an error and share feedback for every learner on every problem solved?

One of the strategies to ensure students make less errors and arrive at correct solutions is that they solve such problems step-by-step and make their reasoning visible. It then helps to understand the source of the error demonstrated through an incorrect step. It can pave a way to correct the error immediately. It is desired that students work out their solutions to a math problem in detail with each intermediate step especially from Grade 6 onwards. For e.g., in a Math problem involving a linear equation 4(4p – 9) = 7(8p + 5), a student is expected to write steps and work out a detailed solution as follows.

Students may commit errors. An erroneous step will eventually lead to an incorrect final answer, and without a feedback mechanism, a student will be deprived of an opportunity that helps him realize which step he made the error and how to correct it.

Students by Grade 8 are expected to solve more complex linear equations for example the following.

Solve the equation:

One incorrect step will eventually lead to wrong answer. How easy is it to comprehend student’s work and find out the step where the error appeared first? Pin-pointing where the error is, what is the error and help overcome them is one of the important learning needs of a learner. It is not practical to have a math teacher/expert give such feedback to every learner for every such problem. For a class of 30 students working on 20 different algebra problems in a worksheet or from a textbook exercise, a teacher needs to evaluate 600 solutions every day and give corrective feedback, which is extremely time-consuming and effort-intensive task. If she needs to give real-time and quick feedback, she needs to check these solutions in her classroom while students work out these problems, which is simply impossible to do every day. What if technology is applied to solve this problem?

The step-by-step solver in Mindspark addresses this aspect of teaching-learning and leverages technology to give personalized instantaneous feedback.

### What is the step-by-step solver in Mindspark and how it uses technology for learning?

Typically, any learning software does either or both of the following:

1. Allows a learner to enter a math problem and then show a detailed solution with each intermediate step leading to a solution, as seen in the figure 1.
2. Poses a math problem, accepts only the final answer, evaluates and show the detailed solution.

Traditional intelligent tutoring systems (learning software) expect a learner to only enter the final answer in a Math problem and then provide feedback if the solution is correct or not. Hardly any of them allows a learner to enter their intermediate steps, get instant feedback on the correctness of the step entered, receive a prompt to facilitate tracing of the error and allow correcting the error in the next step, as per the knowledge of the author.

The Step-by-Step Solver[1] in Mindspark not only evaluates the final answer but also evaluates every intermediate step. It gives feedback instantly when students work out each step of a solution. It thereby helps overcome the practical challenges discussed in the previous section.

In a nutshell, the Step-by-Step Solver in Mindspark,

1. allows the learner to enter the steps of the solution.
2. evaluates each step and gives instant feedback.
3. in case of an error in a step, prompts the learner and gives an appropriate message. This would nudge a learner to arrive at a correct solution eventually. This way a learner learns through feedback and solving problems.

### How does Step-by-Step Solver in Mindspark work?

The Step-by-Step Solver allows a learner to think of intermediate steps and enter them simultaneously while solving a math problem instead of working out in a physical notebook and then entering the final answer. Most significantly, a learner receives instantaneous feedback if the step is incorrect, an explanation of the error, a hint through the message specific to the error, and thus learns to arrive at a correct solution.

After entering the last step, she can click on ‘Submit final answer’ button for the final evaluation (refer figure 6). The Step-by-Step Solver saves the solution and gives feedback if it was correct or incorrect (refer figures 6 and 7). It also shows an expected solution with explanation on some of the key steps of the solution (figure 7). This allows a student understand the expected method(s) and the reasoning especially in case she has worked out a wrong solution. This is similar to what a teacher might also do in a classroom solving a problem on a board.

By allowing a learner to enter intermediate steps, providing instant feedback on correctness for  each step, showing where the error is, and generating customized feedback, the Step-by-Step Solver  handholds students to the correct steps and eventually towards the correct solution. This way everyone learns by answering questions rather than passively showing solutions to mathematics problems.

Error messages in the step-by-step solver are of different kinds. Some may just pinpoint where the error is. Some may also include a prompt to help correct the step as shown in figure 8. Some may be general message like “Check the error, correct and write the step.”

Evaluation of student steps and the solution:

There could be multiple ways for solving a problem. Different ways of answering a problem are considered and any correct solution is evaluated correctly by the step-by-step solver.
Illustration: All the solutions of the problem ‘Simplify: 7v – 2(6v + 6z)’ mentioned below in figures 9a to 9f are evaluated as correct.

There are about 900 questions in Mindspark using the Step-by-Step Solver across diverse content areas such as Algebra, Fractions, Rational numbers, Exponents and Trigonometry.

### How does the step-by-step solver in Mindspark enable learning?

1. As a means of guided practice with instant feedback on each step and the final solution as discussed earlier.
2. As a means to assess learning: It allows a learner to work out solution step-by-step without instant feedback on intermediate steps as seen in the figure below.

Here all the steps in figure 10 are incorrect. The step-by-step solver does not give feedback if the step is right or wrong. It does not generate any error message in this mode.

3. Scaffolding tool to teach (if enabled in a question): It provides scaffolding in 2 different ways as shown below.

• In this mode one of the key steps is already filled and then a student is asked to complete the solution as seen in the figure 11 below.

figure 11: correct step already displayed and a student is asked to complete the solution

• In another mode, the pre-decided step after a particular step is displayed if a student enters say two consecutive wrong steps after the particular step. Such a step is marked as “System has filled this step” as seen in the figure 12 below.

4. Sharing an expected solution(s): The correct solution is displayed using Step-by-Step Solver tool. It can display 1 step at a time with reasoning for each step as seen in this video clip.

As seen in the figure 13 below, reasoning behind each key step (e.g. transposing -36 to RHS and 56p to LHS) is mentioned to the right of the step.

figure 13: steps appear on screen one after another along with reasoning behind each key step

The speed at which each step appears can be customized. (It is kept faster for easier problems and slower for complex problems to give appropriate time for a student to understand.) Because of this, a learner is not overwhelmed with lot of steps on the screen while trying to make sense of the solution.

### What are the potential benefits of step-by-step solver in Mindspark?

1. Writing the entire detailed solution ensures that the learners are thinking through and solving the entire Math problem making their reasoning visible. (Working out a detailed solution with all intermediate steps is an important learning requirement for mathematics especially in grades 7-10.)
2. Each step entered is evaluated and an appropriate error message is shown for an erroneous step. This facilitates learners to learn from the feedback without the availability of a math expert. Learners can correct their mistakes and overcome their erroneous thinking. They can thus enhance their understanding of a procedure and also unleash an opportunity to learn from their mistakes. When a student receives feedback on an error from her instructor in front of a few other students in a classroom, she may feel embarrassed even though not intended by her instructor. This may not be the case when a student solves problems involving step-by-step solver in Mindspark as she is learning at her own pace and receives customized feedback on her screen only.
3. Frequently occurring error patterns observed in students’ solutions in Mindspark when shared with instructors can also help her modify her instructions and facilitate learning in her class eventually. She need not evaluate many students’ solutions for the same.

__________

[1] Ei ASSET is a skill-based test that measures students’ conceptual understanding and benchmarks the school’s performance at international, national & regional levels with actionable insights through easy-to-understand reports.

[2] Block, J. H., & Burns, R. B. (1977). Mastery learning. In L. E. Shulman (Ed.), Review of Research in Education (pp. 16-32). Itasco: Peacock.

[3] Burrows, C., & Okey, J. R. (1979). The effects of a mastery learning strategy on achievement. Journal of Research in Science Teaching, 16(1), 33-37. http://dx.doi.org/10.1002/tea.3660160106

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### Maulik Shah

Maulik heads the maths team of pedagogy research at Educational Initiatives