The changes to courses included eliminating the modern physics course, the second quarter of the electronics course, and the mathematical methods and classical mechanics capstone courses, reconfiguring the junior-year paradigms in physics courses from 9 three-week courses into 6 five-week courses, realigning the computational courses with the revised junior-level paradigms in physics courses, embedding mathematics instruction within all of the junior-level paradigms courses, and developing two new sophomore courses to bridge between the introductory physics courses and the paradigms in physics courses. Still needing discussion and possible revision were the remaining senior level capstone courses as well as possible development of an advanced laboratory course and specialty courses. Faculty involved in implementing the new plan offered the following advice for others implementing similar curricular reforms.
The Paradigms 2.0 Committee identified the following steps to embed just-in-time mathematics instruction in upper-level science courses
sequence required science courses in order of increasing mathematical sophistication
offer “math bits” to provide just-in-time mathematics instruction for understanding upcoming science content
design “math bits” to help students generalize the mathematics, to see how a technique might apply to another situation or topic in an earlier or upcoming science course
design “math bits” to help students recognize similarities and differences among mathematics language and usage as well as science language and usage for similar topics
The Paradigms 2.0 Committee identified the following steps to develop new sophomore courses to bridge between introductory and upper-level courses
Identify issues that need attention such as an overload for students as they begin the upper-level courses, ways to attract and support diverse students in the major, a gap between expectations in introductory and upper-level courses, and/or dissatisfaction with existing courses
Survey students and faculty to gather information about their views
Provide incentives for interested faculty to design a new course
Focus on fundamental science concepts and reasoning skills that strengthen students’ preparation for upper level courses
Use active engagement strategies
Distinguish between the mathematics needed to graduate as a science major and that needed for graduate study in the discipline
include mathematics needed by all majors to succeed in required science courses
encourage students ready for advanced topics, especially those headed for graduate school, to enroll in the graduate level mathematical methods course in the discipline
Assign a separate instructor for teaching “math bits” within a science course
assign an experienced faculty member who has taught many of the courses and has a detailed grasp of the content and mathematics needed in the upper division curriculum
assign a senior faculty member who has strong collaborative and mentoring skills
assign a senior faculty member who can be flexible but also firm if needed to maintain the time and independence necessary to preserve embedding “math bits” in the courses
assign a faculty member who enjoys mathematics and can communicate ways that learning mathematics while learning science enhances understanding both disciplines.
Maintain a commitment to generalizing the mathematics
advocate for just-in-time mathematics instruction rather than using this time for additional science applications or more science content.
explain the importance of students being able to see the math in its general form so they can see where they are using the same math in multiple contexts, sometimes with a variety of notations and different language
make the need evident by engaging students interactively so the primary instructor becomes aware of their struggles and the need to address these difficulties directly
recognize that faculty are used to teaching their own courses in their own ways and may find it difficult to give instructional time to someone to teach mathematics separately
Recognize that “math bits” scheduling needs are different for different courses. For example:
offer in two pieces, one early on and one that comes later with more advanced materials
begin with the science content, even if only for a few instructional hours, to provide context and to help students recognize how the “math bits” they are learning will be useful
perhaps offer as a one week interlude in the middle of a course
Plan collaboratively with the primary instructor of the science course
well before the course begins, arrange for on-going discussions in which the “math bits” instructor and the primary instructor achieve and articulate a common vision for the course
recognize that these instructors may differ on what constitutes the science, the relevant mathematics, and the importance of introducing students to a generalized form of the mathematics
outline what the science content will be and what the relevant “math bits” might be
group together “math bits” that will take more than a day to teach within at most two sets of math sessions
identify any isolated “math bits” that the primary instructor should teach along with the associated science content
recognize that faculty travel, research priorities, family commitments, and even just comfort with flexibility may affect the primary instructor’s available time and interest in such collaborative planning
discuss the primary instructor’s role during “math bits” sessions: best if primary instructor stays in the classroom and is aware of and can build upon the “math bits” instructor’s approach, language, notation, and content
Be both flexible and firm with colleagues
Establish an effective routine for sharing homework assignments.
Respect time issues
Listen and respond to the primary instructor’s requests when feasible
Inform faculty about “math bits” plans and engage them in talking substantively about mathematics content during some upper division curriculum meetings
Enjoy learning new mathematics yourself
Motivate students by connecting fundamental science with current societal issues
Create a coherent narrative
plan so that what students learn in the first week helps in the ninth week and makes them excited to stay on in the course and everything inbetween works like that as well.
look at your plans from a lot of different angles before presenting something to the students to see what the repercussions are going to be if you try to teach it in a certain way
keep the contemporary challenges on the forefront and only do particular science topics when you can link them to such a challenge
Gather and synthesize information from diverse sources
assemble and review relevant books, YouTube videos, and Internet resources
identify Internet data sources that students can access and use to make sense of the world
choose data relevant to the students’ location and interests when feasible
use websites in class to display aspects of a topic visually
access Internet resources to teach yourself as needed
Match the level of instruction to students’ capabilities
become aware of notation and language used in introductory courses
clarify goals, maintain focus, and avoid tangents and unnecessary details
begin with simpler models to build toward comprehension of more abstract models
expect differences in background knowledge and interest in abstract approaches
perhaps coach more advanced students to assume roles as facilitator in their small groups
Create homework sets that emphasize “thinking like a scientist”
keep homework in sync with the topics discussed in class
design problems that require more than simply manipulating numbers
require students to explain their reasoning
choose open-ended aspects of a problem carefully
discuss explicitly the intent of putting open-ended problems on the homework such as the importance of developing understanding rather than simply seeking a recipe to solve a problem
use homework as a context for developing some of the skills needed for analyzing experimental data
create problems that use motivating contexts from everyday life or intriguing situations
emphasize that to do well on exams, particularly in getting started on solving a problem on one’s own, one needs to develop deep understanding through thinking about the homework problems oneself before and after discussing them thoroughly with others, as well as writing up solutions independently
provide model solutions but encourage students not to look for these before working on the problems first themselves
Integrate laboratory experiences within the course
try out and modify the experiment oneself as needed before using in class
include opportunities for students to be thoughtful and creative in this context
monitor and assist small groups as needed
perhaps introduce roles such as “driver” and “navigator” to be sure that all students have opportunities to handle the equipment
clarify the goal, whether testing a hypothesis or just exploring relationships among quantities
provide access to the equipment outside of class for those who need extra time
be clear about expectations for write-ups
wrap-up by discussing findings and their implications with the whole group
note difficulties and plan ways to address these in earlier class sessions next time
Design intriguing demonstrations
use everyday examples of phenomena similar to aspects of the more complex phenomena to be discussed
incorporate familiar technology such as cell phones if feasible
include students as participants when possible
Foster student engagement during class sessions
welcome student questions
become aware of and begin using interactive engagement strategies
pose issues that interest students
plan explicitly for one or more small group activities in every class session
make clear connections between activity and contemporary challenge
Communicate clear guidelines for a term paper assignment
discuss expectations in class, welcome questions during office hours, provide written directions
explain goal of presenting multiple perspectives explicating both the science and issues related to the world
encourage students to seek feedback on topic before investing a lot of time
provide details of requirements for graphs, diagrams, and explanations
value a student’s thoughtful writing (even if math mistakes affect conclusions)
Seek student feedback in multiple ways
encourage students to come to office hours
listen to small groups during in-class activities and encourage questions
encourage student questions in class
Use technology to enhance instruction
Think about three kinds of goals: science content goals, math content goals, and sense-making goals. Consider:
how these goals are related to each other
how to interleave these goals in sensible ways
how to make the sense-making goals central and not peripheral
Think about how to embed sense-making goals in all aspects of the course. Consider:
how to make sense-making goals explicit in class, during both small group work and interactive lectures
how to request sense-making as an expected part of problem-solving in homework
how to create the expectation of sense-making as required in exams.
how to foster sense making at the beginning in orienting oneself to the problem situation as well as at the end in evaluating an answer, how understand it and have confidence that it is correct.
Think about how to assess sense-making skills at the beginning and end of the term
Think about how to scaffold and then fade instruction in sense making
early in course be explicit about requesting sense-making in class, on homework, and the first exam, such as “Consider special cases: Does your result for the maximum range of a projectile on an incline make sense if the ground is horizontal? If the ground is vertical (like right up against a cliff)?”
state explicitly that credit will be given on an exam for sense making about an answer, about why an answer is correct; even if just realize something is wrong and explain why know it is wrong
in the middle of the course, fade to a more general format such as “utilize at least two sense-making strategies to evaluate an answer”
near the end of the course, fade to an even more general format such as “Be sure to do some sense-making around your result.”
Think about how to build on prior courses and prepare for later courses
Think about what content to cover when combining multiple courses into one. Consider:
what content is going to be relied upon in subsequent courses
what content is motivating for students
what combination of content makes a coherent story
what content can be omitted
Do a lot of thinking about what will happen during Week 1, because this first week sets the tone for the rest of the course
focus on thinking about how to engage students in doing physics rather than on how to convey specific physics content.
gain the student perspective at this early stage by involving graduate students, postdocs, and/or advanced undergraduates in the conversation.
draw on sense-making strategies discussed in the research literature, such as asking small groups: What are you doing? Why are you doing it? How will it be helpful? (p. 63) in Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan
plan to keep small groups small, groups of 2, or at most groups of 3
be aware that the design of a classroom affects class dynamics; what may be possible in a
studio classroom differs from what can be attempted in a traditional lecture hall.
During first week of class, do a variety of active engagement strategies to set the tone for the rest of the term
make sure there are a lot of student voices during each class
learn student names quickly and encourage students to speak up
carry out an interactive lecture by, for example,
asking students lots of questions to find out what they know and are thinking about a topic
inviting students to say what the next step is when working out a problem on the board
welcoming students’ questions as contributions useful for all to consider in the midst of collaborative thinking
facilitate a whole group discussion in which many different students contribute their ideas
foster participation by asking students to talk briefly with their group members about a particular issue and then for the small groups to share their thinking with the whole group
ask
small white board questions to which students respond individually in writing, even if only with a ? mark if they are feeling bewildered; then use some of the responses in various interactive ways.
get students up out of their chairs by doing a
kinesthetic activity in which students use their own bodies to help them visualize a physical situation.
do a
compare and contrast activity in which small groups work, for example, on slightly different versions of the same calculation and participate in a wrap-up discussion about what the different results mean.
do a
small group activity in which, for example, students work on a problem, use a computer simulation, or explore phenomena.
when students are working on large whiteboards in small groups, make sure everyone has a pen, everyone has an eraser, everyone has access to the white board and is contributing to the thinking
during a small group activity, bring the whole group together several times briefly to provide guidance as groups encounter expected issues as they work.
have small groups work on large whiteboards and invite groups to present and discuss their work with the whole group
be supportive and positive as students become accustomed to what for many may be an unusual culture if they are used to sitting quietly during lectures.
During first day of class, attend to administrative details related to using active engagement strategies
arrange for an assistant to take pictures of the students so you can learn their names
include time for downloading on their computers any software needed for small group work and homework, such as Mathematica
remember to ask students to pick up small whiteboards, pens, erasers, and large white board for their group as they enter the classroom so they already have these materials at hand when need to use them in later classes
remember to inform students of department resources for fostering a sense of community among majors, such as a room where they can study and work together on homework
Be aware of and plan ways to mitigate as well as utilize a wide range of experience and preparation among the students, particularly if some are still completing the introductory series and others are transfer students already enrolled in upper-level courses.
ask department advisor to make sure students have course pre-requisites before they enroll; clearly state course pre-requisites on the first day of class
make a list of prerequisite knowledge and skills that students ought to have for undertaking the topics and activities planned; consider where students might have learned this and how to provide for those who have not
perhaps create a visual display with post-it notes in one’s office, as a reminder of the many aspects that need to be addressed
choose an initial problem in a familiar context, a challenging problem but one for which even the least prepared students likely have resources for doing the kind of sense-making intended
acknowledge the wide range of experience and preparation within the class; encourage those who hear a word or idea expressed that they do not understand to ask immediately for clarification
encourage or arrange for the more advanced students to be spread out among the small groups rather than clustered within one or a few groups
invite the more advanced students to learn deeply by teaching others, as informal mentors within their small groups
make clear that all members of a group need to understand a problem’s solution, that the success of a group depends on every member being able to report and discuss the group’s solution
make explicit the expectation that all members of a group will stand up and participate in the group’s presentation to the class and that any member of the group may be called upon to explain some aspect of the group’s solution
Focus upon the reasoning part of a solution during small group problem-solving activities
Invite small groups to report out their answers and to talk about strategies they used to look for mistakes as well as to build confidence in their answers
ask each group to discuss one thing they did to evaluate their final answer
make a list on the board of the strategies students mention
identify this list as an initial sense-making framework for use in class, on homework and exams
Early in the course, include explicit sense-making prompts on worksheets setting up small group problem-solving activities in class and on homework. Ask students, for example:
to consider special cases
to plot a function and to physically interpret its shape
to compare an answer with an expectation based on prior knowledge or everyday experience
to analyze if and how the answer depends upon certain physical quantities
to check that units and dimensions are the same on both sides of equations
Be sure grading of the homework provides feedback on the sense-making aspects of a problem.
Listen closely to what students are saying in their small groups and during class discussions to learn more about how they are thinking
if a problem is more challenging than anticipated, take more time than initially planned to explore its nuances
invite small groups to come up to the board and do the solution, to write it large enough for all to see, to talk about each step, and to welcome questions from other students
make clear that it is the responsibility of the other students to ask questions and to express their confusion even if they are so lost they are not even able to articulate a question
if a student makes a mistake, wait to see if it is noticed by other students, and commend a thoughtful discussion among the students that addresses the issue without embarrassing the student presenter
if a student presents a novel correct solution, discuss the similarities and differences among productive strategies
When working a problem on the board, ask students what to do next
encourage students to think about how to get started on a problem
welcome student contributions even if incorrect
try to elicit common stumbling blocks and discuss them explicitly
elaborate on suggestions that seem to need more explanation
During office hours, engage students in working problems together; be a resource for them as they help one another rather than doing all the talking
enjoy and learn from watching students problem-solve “in the wild”
ask questions to help students develop new understandings rather than telling answers
affirm that lots of practice helps in working lengthy problems written symbolically
use a variety of sense-making prompts, some specific to the context, some more general, in interacting with the students
Be aware of language issues for students who are not native speakers
write sentences on the board that mirror what you are saying
make sure they have access to the textbook
encourage students to come to office hours
create opportunities where they can feel comfortable, have more time, and less pressure so are more able to formulate some questions
Meet before each class with assistant(s) to discuss their impressions of the previous class, plan the upcoming class, and discuss ways to improve those plans
rehearse plans for small group activities and interactive lectures
discuss likely student difficulties
listen to informal feedback about student experiences in the course
After a few weeks in the course, ask for direct student feedback with an anonymous survey
ask how difficult the course seems with a Likert scale from very easy to about right to very difficult
ask how much time students are spending on the homework and encourage those who seem to be spending too much time on it to come to office hours and get help by asking questions
ask what if anything students like about the course * ask what if anything students would like to change about the course
As course progresses, consider whether choices of subject matter and/or resources to include are appropriate and whether topics thought to be important for later courses are really necessary to be addressed in this one
if a topic seems to take more time than expected, reconsider its importance in the course
consult with colleagues if contemplating dropping a topic that would have prepared students for a later course
consider wrapping up a topic by ‘telling’ information needed if students are not making enough progress on their own
choose not to use software such as Mathematica in class if it seems too distracting; limit use to homework with help offered in office hours as needed
choose to continue specific sense-making prompts in contexts where the subject matter calls for a different kind of sense-making than used earlier.
Recognize one’s own limitations in launching a new complex endeavor
when discussing a challenging problem with lots of parts, recognize that the cognitive load of keeping in mind what needs discussing may limit your ability to engage with students to see what their ideas are
be aware that it is hard to predict a reasonable schedule when teaching interactively; expect to readjust plans as find out how students are having trouble and what their questions are
acknowledge that it is hard to respond in the moment to students who are struggling when encountering their difficulties for the first time
accept that teaching relatively new content means multiple stresses, in making sure one is correct in what one is teaching, in devising elegant ways to say things clearly, in becoming aware of and responding to student difficulties, and in creating a coherent narrative where there is a strong progression of ideas and everything fits well together.
acknowledge the tension between recognizing that not every student is going to understand everything in the course and caring about every student, wanting to help every one of them.
Reflect often and keep a record for oneself
write briefly about each class session
create a binder with session plans, handouts, and commentary that will help one remember what one wants to change during the next version of the course
debrief frequently with a more experienced faculty member to gain perspective and insights about these challenging ways of teaching and learning