I also add the caveat that no one intentionally wants the following outcomes. They are unintended consequences, and they are consequences we should know about. Here's a partial list of negative attitudes or beliefs that has been documented. I've mentioned these before.

- Memorizing facts and formulas and practicing procedures are sufficient to learn mathematics.
- Mathematics textbook problems can only be solved using the methods described in the textbook.
- Teachers and textbooks are the mathematical authorities.
- School mathematics is driven by rules and memorization, and is driven by procedures rather than concepts.
- If a problem takes longer than 5-10 minutes, then there is something wrong with the student or the problem.
- The goal of mathematics is to obtain one correct answer and do it quickly.
- The teacher is the only source of determining whether an answer is correct or incorrect.
- Students’ role in the classroom is to receive knowledge by paying attention in class and to demonstrate it has been received by producing right answers.
- The teacher’s role is to transmit knowledge and verify that it has been transmitted.
- Only geniuses have what it takes to be good at mathematics.
- Students prefer to have only one way of solving a problem, because it is less to memorize.
- The processes of formal mathematics have little or nothing to do with discovery or invention.
- Students who understand mathematics can solve assigned problems in 5 minutes or less.
- One succeeds in school mathematics by performing the tasks, to the letter, as described by the teacher.
- The various components of mathematics are unrelated.

So when an instructor teaches via IBL and works with students who have

*never*had an IBL experience, there usually exists a host of default expectations that one has to work through with the students.
Let's say the students in your class are reluctant to buy into the IBL system. Then looking at teaching as a system can provide a broader perspective to address the issues, which could lead to a more coordinated effort to get students to become active participants in their own education.

Attack this teaching challenge through content, assessment, and pedagogy (i.e. through a system approach).

*Content*

Course content is one of the key components. If tasks are too hard and only the very best students are successful at solving them, then students who are more likely to have some of the negative beliefs and attitudes above will have "evidence" to reinforce them. "I'm not going to be successful anyways, so I might as well give up after 5 minutes."

Consequently, the tasks presented to students should be matched to their levels, and having a handful of very accessible problems is almost never a mistake. At worst, the students mow them down, and you've been able to get more students to participate.

If students are having trouble starting up or you have a subgroup that is passive and not engaged, then one tactic available is to breakdown a problem set (or a part of one) into more manageable pieces. This method can help students get going on problems. One thing to keep in mind is to keep problems that are more challenging to keep everyone engaged at an appropriate level.

Consequently, the tasks presented to students should be matched to their levels, and having a handful of very accessible problems is almost never a mistake. At worst, the students mow them down, and you've been able to get more students to participate.

If students are having trouble starting up or you have a subgroup that is passive and not engaged, then one tactic available is to breakdown a problem set (or a part of one) into more manageable pieces. This method can help students get going on problems. One thing to keep in mind is to keep problems that are more challenging to keep everyone engaged at an appropriate level.

*Assessment*

If assessment is setup only along traditional lines (homework, midterm, final), and assuming each of these components is implemented in the usual way, then the ethos of the IBL course is mismatched to the course assessment. If homework is graded based on accuracy, and there are no venues for safe exploration, then students are being told implicitly that "mistakes count against you." This can inhibit risk taking and undermine the course. While our viewpoint from the instructor side is that we expect the exploration to take place on scratch paper, this may not be a concept and workflow practiced by students. It's a curious thing -- students expect to write down the answer in one shot. This is absurd in art, science, math, etc. when thinking about it. So how did we get here?

The pristine nature of clear lectures can be beguiling. The imagery is one of a the brilliant expert doing it right the first time. Every time. Instructors are never seen struggling, and this is an impossible standard to achieve as a learner, especially for the ones who need the most help. The unintended consequence is that in our effort to make the best possible presentations, the notions that working hard, being willing to explore, learning from mistakes, and having the predilection to work through our own personal learning obstacles can be undermined.

Homework is an opportunity to portray doing math the way mathematicians do it. One option is to provide feedback but not grade the homework in the usual way with numerical scores. The explicit message that should be given to students is that their effort and the quality of their exploration is being checked, and feedback will be given to them to assist them with the learning process. I'm not saying that this is how it should always be done, but this is an idea worth considering and adapting for your own courses. If your goal is to bring about change in student perceptions in math, then grading homework for process and effort is one of the available tools.

Grading presentations is valuable. In courses where presentations are used regularly, presentation grades should be reflected in final course grades. Students are being asked to share their ideas, a noble thing indeed, and the quality of this work should be reflected in their course grades. A level around 25% works well in lower-level courses. As courses become more proof oriented, then raising the presentation grade as a percentage of the overall grade makes sense. (Grading presentations and some sample rubrics will be discussed in future posts.)

Mastery-based finals are used by some IBL instructors with success. Such exams are split into two parts. Part 1 consists of material that anyone who passes the course should know. Part 1 is contains the fundamentals and essential basics. Students should be able to complete the entirety of part 1. Part 2 is an opportunity for students to raise their grades. Part 2 contains problems that require applying knowledge to problems novel to the students. (Parts 1 and 2 are given to students at the usual examination time.) A final exam structured like this allows students opportunities to demonstrate that they have learned the material and can demonstrate that they can use what they learned to solve problems new (to them). Before the final exam, students are given a (non-final) course grade going into the final exam. Passing part 1 means students keep their grade OR get at least a C- in the course. Success on part 2 only improves the final grade and does not count against students (i.e. non-negative help).

Why do mastery-based final exams make sense? If we value learning, applying ideas, and self-improvement, then mastery-based finals are a form of assessment of mathematics aligned to these values. Further it provides an incentive for students to work hard through the end of the course, because they have a chance to succeed right up to the very end. It is also noted that this does not mean one lowers standards. The idea here is to keep the standards at the usual level, but provides a structure and incentives for students to achieve these standards.

The three main examples (effort/process based homework, assessing presentations, and mastery-based finals) are ways we can align assessment of the usual assessment items to IBL classes. Additionally instructors can use portfolios, reading assignments, and reflective writing (journaling) as additional forms of assessment. These assignments do not need to be large part of the final course grade, and are valuable tools for gathering data about how your students think. Gathering formative assessment is significantly valuable, and all instructors are encouraged to use one or more of these additional strategies to help put together the full picture.

A specific example I want to share is journal assignments. Specifically this example is about using "The 5 Elements of Effective Thinking," by Burger and Starbird as a vehicle for students to think about their own thinking (metacognition). When working with a group of students who do not like mathematics (e.g. Math for Liberal Arts), then having students read and write about healthy ways of doing math is a successful strategy for combating math anxiety and buy-in. Math autobiographies are normally the first assignment, and the rest of the assignments are based on chapters from the book. A typical assignment asks students to (a) read the assigned chapters, (b) write about 2-3 things they learned, and (c) write a personal reflection about how they are doing math.

For students in education courses, such as "Math for Elementary Teaching," I use a variety of articles from Teaching Children Mathematics, and books like "What's the Point of School?" by Guy Claxton, and "What's Math Got to Do With It?" by Jo Boaler. Students in math major courses could read autobiographies of famous mathematicians, books like "Fermat's Enigma" by Simon Singh, or other related books. A vast array of possibilities exists with journal assignments, and you are encouraged to find (and share!) strategies that work for you.

One last facet of assessment is that it needs to be set before the term starts. Assessment is the least flexible component, once the term gets rolling. Thus, it is worth thinking about assessment early in the planning phase of a course well before you meet classes for the first time.

The pristine nature of clear lectures can be beguiling. The imagery is one of a the brilliant expert doing it right the first time. Every time. Instructors are never seen struggling, and this is an impossible standard to achieve as a learner, especially for the ones who need the most help. The unintended consequence is that in our effort to make the best possible presentations, the notions that working hard, being willing to explore, learning from mistakes, and having the predilection to work through our own personal learning obstacles can be undermined.

Homework is an opportunity to portray doing math the way mathematicians do it. One option is to provide feedback but not grade the homework in the usual way with numerical scores. The explicit message that should be given to students is that their effort and the quality of their exploration is being checked, and feedback will be given to them to assist them with the learning process. I'm not saying that this is how it should always be done, but this is an idea worth considering and adapting for your own courses. If your goal is to bring about change in student perceptions in math, then grading homework for process and effort is one of the available tools.

Grading presentations is valuable. In courses where presentations are used regularly, presentation grades should be reflected in final course grades. Students are being asked to share their ideas, a noble thing indeed, and the quality of this work should be reflected in their course grades. A level around 25% works well in lower-level courses. As courses become more proof oriented, then raising the presentation grade as a percentage of the overall grade makes sense. (Grading presentations and some sample rubrics will be discussed in future posts.)

Mastery-based finals are used by some IBL instructors with success. Such exams are split into two parts. Part 1 consists of material that anyone who passes the course should know. Part 1 is contains the fundamentals and essential basics. Students should be able to complete the entirety of part 1. Part 2 is an opportunity for students to raise their grades. Part 2 contains problems that require applying knowledge to problems novel to the students. (Parts 1 and 2 are given to students at the usual examination time.) A final exam structured like this allows students opportunities to demonstrate that they have learned the material and can demonstrate that they can use what they learned to solve problems new (to them). Before the final exam, students are given a (non-final) course grade going into the final exam. Passing part 1 means students keep their grade OR get at least a C- in the course. Success on part 2 only improves the final grade and does not count against students (i.e. non-negative help).

Why do mastery-based final exams make sense? If we value learning, applying ideas, and self-improvement, then mastery-based finals are a form of assessment of mathematics aligned to these values. Further it provides an incentive for students to work hard through the end of the course, because they have a chance to succeed right up to the very end. It is also noted that this does not mean one lowers standards. The idea here is to keep the standards at the usual level, but provides a structure and incentives for students to achieve these standards.

The three main examples (effort/process based homework, assessing presentations, and mastery-based finals) are ways we can align assessment of the usual assessment items to IBL classes. Additionally instructors can use portfolios, reading assignments, and reflective writing (journaling) as additional forms of assessment. These assignments do not need to be large part of the final course grade, and are valuable tools for gathering data about how your students think. Gathering formative assessment is significantly valuable, and all instructors are encouraged to use one or more of these additional strategies to help put together the full picture.

A specific example I want to share is journal assignments. Specifically this example is about using "The 5 Elements of Effective Thinking," by Burger and Starbird as a vehicle for students to think about their own thinking (metacognition). When working with a group of students who do not like mathematics (e.g. Math for Liberal Arts), then having students read and write about healthy ways of doing math is a successful strategy for combating math anxiety and buy-in. Math autobiographies are normally the first assignment, and the rest of the assignments are based on chapters from the book. A typical assignment asks students to (a) read the assigned chapters, (b) write about 2-3 things they learned, and (c) write a personal reflection about how they are doing math.

For students in education courses, such as "Math for Elementary Teaching," I use a variety of articles from Teaching Children Mathematics, and books like "What's the Point of School?" by Guy Claxton, and "What's Math Got to Do With It?" by Jo Boaler. Students in math major courses could read autobiographies of famous mathematicians, books like "Fermat's Enigma" by Simon Singh, or other related books. A vast array of possibilities exists with journal assignments, and you are encouraged to find (and share!) strategies that work for you.

One last facet of assessment is that it needs to be set before the term starts. Assessment is the least flexible component, once the term gets rolling. Thus, it is worth thinking about assessment early in the planning phase of a course well before you meet classes for the first time.

*Coaching*

As mentioned in an earlier post (Link), coaching is an critical facet of teaching. When students get stuck, then your role as an IBL instructor is to manage the classroom so that students continue to learn, as opposed to giving up. If the default attitude or belief of your students is to shutdown when stuck, then coaching students through this phase is part of the job.

In summary, a "system approach" to dealing with a complex issue (students' attitudes about Math) provides a richer and likely a more successful approach. Teaching and learning is complex, and there are many things that affect the day-to-day activities of our course. At times it can be a daunting challenge, but as my colleague Professor Dylan Retsek says, "Chop wood, carry water." In this case, we break down our response to content, assessment, and coaching.

Upward and onward!

- Coaching can be verbal encouragement. "It's okay to get stuck. Let's see if we can figure this one out together. Don't give up... Being stuck is a natural, regular part of doing mathematics."
- Coaching can be directive. "Let's try figuring this related example/strategy/definition..." or "Let's pair up and try to see if you can think of a useful idea or draft a plan you can take home and try."
- Coaching can be done through content. "Let's look at this supplemental handout I wrote yesterday after class, when I saw that we were all stuck on number 19. I think these problems will be really helpful..."

In summary, a "system approach" to dealing with a complex issue (students' attitudes about Math) provides a richer and likely a more successful approach. Teaching and learning is complex, and there are many things that affect the day-to-day activities of our course. At times it can be a daunting challenge, but as my colleague Professor Dylan Retsek says, "Chop wood, carry water." In this case, we break down our response to content, assessment, and coaching.

Upward and onward!