Category Archives: Learning and teaching

Notes on “Abolitionist Teaching and the Future of Our Schools”

Accessed on June 29, 2020
Bettina Love,
Gholdy Muhammed
Dena Simmons
Brian Jones.

Bettina Love described arriving at college and being pushed onto a “jock track”. She transferred universities, but that was a turning point in her life.

Gholdy Muhammad recalls reading a hadid about ways to respond to oppression. Reading and studying Blackness also helps her hold on to abolitionist teaching.

Dena Simmons was put in the “slow class” in first grade and advocated to switch. Then in high school, leaving the Bronx, she experienced tone policing and hurt. Returning to the Bronx as a teacher, she saw the way the system was set up to fail her students.

Q: (Brian Jones) How does White supremacy show up in schools?
A: (Dena Simmons) Tone policing, tokenization and objectification, over-managing, expectation of being super-people. The emotional labor of asking “am I safe to be Black here” is also work.
(Gholdy Muhammad). It shows up in the curriculum, in the interviewing questions (“how does anti-racism appear in your math curriculum” isn’t one of those questions. The questions focus on meeting “challenges”). It is in tests, in the emphasis on skills, in zero tolerance policies.
(Bettina Love). In education “we manage inequality”. We don’t remove it. Instead of removing barriers, they manage inequity (via directors of equity).

Q: (Brian Jones) White supremacy follow up
A: (Dena Simmons) “We can use anything in education to create harm”. Social-emotional learning (feelings) is often equated to promoting equity. But “if the school is steeped in White supremacy… then the curriculum is a White supremacist curriculum”. Without trauma informed instruction to address the trauma of racism, there is an issue.

Q: (Brian Jones) Abolition is being used about prison and police. What about schools?
A: (Betinna Love) “It is not a radical thing to want to be seen as fully human”. Abolition is a request to start over. “It is a push for everybody’s humanity”. See work by Angela Davis, Ruth Wilson Gilmore. Eradicate the conditions that make it possible to treat children in the way they are treated within school.  The system is oppressive.  Build a community-based curriculum where children learn their history and culture, have the chance to play, and have access to healing.  When you’re an abolitionist, you might not live to see the win, but you work thoughtfully and methodically, to justice.

(Gholdy Muhammad).  “Stop putting fresh coats of paint on the same debilitating structures”.  State learning standards, teacher evaluations, curriculum.  Who authors these?  In the 19th century, Black literary societies came together to strategize to improve social conditions with four learning goals: identity, skills, intellectualism, criticality (understand power).  Teaching these four goals would teach the whole child.

Q: (Brian Jones).  Black women and Black queer folks are important in this.
A: (Bettina Love) For Black queer women, the intersection creates deep knowledge of marginalization.
(Dena Simmons) There is so much erasure of the contributions of Black womyn.

Q: (Brian Jones). What about the shutdown of schools?
A: (Bettina Love).  They canceled standardized tests, handed out computers, relied on the ingenuity of teachers, asked for flexibility and compassion, and asked parents to be partners.  “Why did it take a pandemic” for this?  Expelling children, school shootings, should not be normal.  When schools reopen, that trust in parents and teachers should remain.
(Dena Simmons). Some students are thriving in the shutdown of schools.  Learn from that.  “The school has never known what’s best for us”.
(Gholdy Muhammad).  Be urgent in pedagogy.  Skills and knowledge are not enough.  Students need to be able to agitate as antiracists.
(Dena Simmons).  How will you change how you spend money, etc, grounding in anti-racism?
(Bettina Love).  The work needs to happen in White schools, including anti-racist education.
(Gholdy Muhammad).  Redefining achievement and success is important.  “High performing” schools are not a positive model.
(Bettina Love).  “Black parents care”
(Dena Simmons).  The school can be a place of trauma for the parents.

Q: (Brian Jones) Resources people can turn to?  Black Lives Matter at School movement.  Four demands.  Counselors not cops, hire Black teachers, end zero tolerance discipline, teach Black history and ethnic studies.
A: (Dena Simmons) For “woke” White people, start at home with racist family members.
(Gholdy Muhammad).  See  A goal is to have curricular materials made by Black educators.
(Bettina Love).  There are so many organizations doing the work.  See “Unapologetic” by Charlene Carruthers

Q: (audience Q) Trapped in standardized tests, what do you do on a daily basis?
A: (Dena Simmons).  In one example, as an act of resistant in a 7th grade classroom, talked through with the students about whether to take a standardized test (and the students chose not to).
Community organizing is also an act of resistance.
(Gholdy Muhammad).  “Revise and modify the curriculum that is given to us”.  “How can this unit plan help my students learn about themselves”?  Plan towards advancing student thinking about equity, power, anti-oppression.  “Collect the data you want to collect”.  “If you value it, assess it”.

Q: Unions?
A: (Dena Simmons) Let’s ask our unions to demand these practices.

Q: Do we need to create our own schools?
A: (Gholdy Muhammad).  “Black abolitionists… cultivated their own schools”.  “A lot of power lies with the parents”.
(Bettina Love).  “It is our money”.  We pay the police, teachers, etc.  “Understand the value and power we have as citizens”.

Q: Is abolitionist teaching anti-capitalist?
A: (Bettina Love).  Abolition is anti-capitalist.
(Dena Simmons). The country is “built on stolen Black labor”.

Notes on: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”

SE Anderson, 1990: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”.  The Journal of Negro Education

Section one: “A few grim statistics”

The manuscript starts with “A few grim statistics”, looking forward to 2000 and to 2010 and providing numbers about low numbers of Black, Latino, and Native American scholars receiving PhDs in mathematics, physics, and astronomy in the 1980s.

Section two: “The Eurocentric Basis of Mathematics”

The author writes that educators have a duty to build a system “based on the assumption that any person can learn anything”, with the “true beauty of mathematics” coming from the creation of “logical systems that help explain the complexities of Nature”.  (See Joseph 1987 p 22-26).


  1. math is often specifically delinked from materialist concerns
  2. math is confined to an elite group with special gifts
  3. math discovery comes from “deductive axiomatic logic” not from “intuitive or empirical methods”
  4. math results needs to be presented in a particular style, so “new additions to mathematical knowledge” are from a small, special, Eurocentric group.

The narrative of mathematics is presented as going from the Greeks through “European men and their North American descendants”.  “African, Indian, Chinese, or Mayan contributions” receive short mentions in textbooks.

In addition, an unappealing curriculum ends up reinforcing “racist assumptions about people of color”.

Section three: “Six Pedagogical Disasters in Mathematics Education”

  1. “separate arithmetic from algebra”
  2. “teach mathematics without any historical references”
  3. “use textbooks that are elitist and cryptic”
  4. “do work and be tested as an individual” (vs in study groups)
  5. “accept the myth that mathematics is pure abstraction”
  6. “memorize”

1: the author asserts that arithmetic is taught first (for many years), followed by algebra: evidently these subjects are kept separate.  I don’t have any thoughts on this.

2: teaching math as ahistorical: a bunch of European names are attached to math facts and abstractions, without the humans themselves being introduced.

3: cryptic texts: these make math appear unaccessible and not for most people.

4: the individual: math classes are often structured competitively / individually.  this is not how people actually solve problems.

5: real math is abstract: “erudition, abstraction, and compartmentalization”, distance people from creative sources and make math seem unnatural, rather than a natural human act.

6: memorization: math problem solving is often turned into the memorization of definitions, theorems, etc.

Section four: “An alternative curriculum and pedagogy”

See Bob Moses and his “algebra project” (learning algebra in elementary school via subway rides).  See Arthur Powell and the “Writing in Math” project.  See Marilyn Frankenstein and “radical math”.  Look for alternatives “to the arithematic-algebra-precalculus-calculus `learning’ sequence that is so pervasive yet so devastating.”  College-level courses should (1) “show the interconnectedness of mathematics and real-world problems” and (2) should “show how people throughout history have created mathematical techniques to solve problems”.

Section five: “How and what I teach”

  1. Via “psychological upliftment”, “emphasizing that ordinary people create mathematical ideas and ‘do’ mathematics”.
  2. By assuming “the role of a confidence builder”.  Letting students know “they all have the intellectual capabilities to understand the material”
  3. Attributing not understanding to “my own or the textbook’s failure to communicate clearly”
  4. Choosing “the quality of mathematics knowledge” over the quantity.  “I may set out to cover six chapters… if they complete only three or four chapters and learn those well, then I am confident they can pick up the rest”

In a typical algebra class: first two classes focus on historical, cultural, sociopolitical.  Relate math to “humanity’s ongoing struggle to understand Nature”.  And to “capitalism’s attempts to control and dominate Nature”.  Emphasize that “some of the very first mathematical/scientific thinkers were African women”.  “Show how early mathematics and science led to the building of the pyramids, the Great Wall of China, and the road to Kathmandu”.  Astronomy, astrology, iron-smelting, surgery, etc.  Name that Euclid “spent 21 years studying and translating mathematical tracts in Egypt” and “Egypt is in Africa and that the people who inhabit the land were and are Africans”.  Pythagoras also studied in Egypt (and perhaps India).  The theorem attributed to him existed 1000 years before him in Babylonian documents.

The “intent… is to shatter the myth that mathematics was or is a `White man’s thing'”.  “I show how certain aspects of European mathematics could not have developed had not the Europeans traded with more advanced societies”.  Example: “The Vatican denounced Hindu-Arabic numerals”.

The research university of Bait al-Hikma is important because of Mohammed ibn-Musa al-Khwarizmi (the name “al-djabr or `algebra’ comes from a text he wrote) and the term “algorithms” is a corruption of his name (dating to his second book, “Algorithmi de Numero Indorum”).  “I also mention… the algebraist Omar Khayyam (c. 1050 – 1122 AD)”.  See Nasir Eddin al-Tusi for non-Euclidean geometry, as well.

In the 1600s, when Europeans were trying to build very large ships to carry African slaves, they used the knowledge of “Gambian, Chinese, and Indian mariners”.  “Developments in hydrodynamics (and its attendant mathematics) contributed … to the horror of .. the slave trade”.

“I further point that that calculus was created to facilitate the study of ballistics” in wars by England and Germany.  Military needs “continue to inspire many mathematicians and scientists to purse the War Machine”.

Structuring class: (1) “About two weeks into a class, I facilitate the creation of study groups” (3-4 people whom students choose).  These groups are used in and out of class.  The groups complete progress reports (no tests).  (2) “I also incorporate a weekly 15- to 20-minute class discussion” of a news article from the Science Times, to emphasize the relationships between math and the social / natural sciences.  (3) students are asked to make notecards of facts that they should bring to class.  (4) computational tools are encouraged.

Section six: “Conclusion”

“a subtle but effective form of educational genocide is taking place”.  “To offer an alternative that is genuinely egalitarian and truthful we must open our eyes to the centrality of the contributions made by the vast majority of the world’s people”.

Python in my dynamical systems class

I have been using Mathematica in my dynamical systems class for a few years. I don’t have a systematic curriculum related to it, though, and need to develop clearer computational learning goals, as well as a pathway for students to develop computational skills.

Ideally, by the end of the semester, students would be able to do an analysis of a one-parameter dynamical system with the aid of computational tools. They would find fixed points, identify stability, create phase portraits and bifurcation diagrams, and perhaps create stability diagrams. I would expect them to be able to identify global bifurcations, as well. For limit cycles, I need to make a decision on my expectations. I suppose I would like for students to be able to create a curve of initial conditions, use “Events” in Mathematica of Matlab integration, and identify the stability / existence of a limit cycle in a 2d system.

At the moment I’ve been relying on Mathematica and some students have chosen to use Matlab. I would like to move towards Python. Currently the dynamical systems course is the only place students work with Mathematica, while Python is an option across a range of courses.

This means I need to learn how to set up Python for a class. At the moment I’m taking a look at Koehler and Kim, 2018 for some guidance on this. They go in the direction of Jupyter Notebooks so I will explore that for now.

1) Install the Anaconda application on my Mac.
2) Open the Anaconda Navigator: it has seven options when I first open it (Jupyter lab, Jupyter notebook, Qt console, Spyder, Glueviz, Orange 3, RStudio). The first four are already installed and I have the option to Launch them. For the other three, I have the option to install them. I’ll launch Jupyter notebook.

Koehler and Kim 2018: Interactive Classrooms with Jupyter and Python. The Mathematics Teacher. Vol. 111, No. 4 (January/February 2018), pp. 304-308 (5 pages)

Notes on “How a Detracked Mathematics Approach Promoted Respect, Responsibility, and High Achievement”

Boaler, 2006.  “How a Detracked Mathematics Approach Promoted Respect, Responsibility, and High Achievement”.   Theory Into Practice, 45:1, 40-46

This article is about a high school math program with high and equitable math achievement, where mixed-ability approaches led to “higher overall attainment and more equitable outcomes”.  The students in this study developed “extremely positive intellectual relations” with peers across culture, social class, gender, and attainment “through a collaborative problem-solving approach”.

The article describes a problem solving approach that was used ad the school and that enables these outcomes.  The problem solving approach (“complex instruction”) involved “additional strategies to make group work successful”.  The author identifies seven factors: “The first four (multidimensional classrooms, student roles, assigning competence, and student responsibility) are recommended in the complex instruction approach; the last three (high expectations, effort over ability, and learning practices) were consonant with the approach and they were important to the high and equitable results that were achieved.”


Ingredients in the approach:

(1) Multidimensionality:  In some classrooms success is about “executing procedures correctly and quickly”.  Here, success requires a range of abilities where “no one student ‘will be good on all these abilities’ and … each student will be ‘good on at least one'”.  Giving students “group-worthy problems”: “open-ended problems that illustrated important mathematical concepts, allowed for multiple representations, and had several possible solution paths (Horn, 2005).”  Students were able to identify: “asking good questions, rephrasing problems, explaining well, being logical, justifying work, considering answers, and using manipulatives” as contributing to success in mathematics.

This breadth was key: that there are multiple paths to an answer, with interaction and explanation central to the work.

(2) Roles: “facilitator, team captain, recorder or reporter, or resource manager”.  If each student has something important to do in the group, they are needed for the group to work.  The teachers reinforced the centrality of each role by pausing to ask facilitators to help with answer checking, etc.  This helps with the reliance of students on each other.

(3) Assigning competence: “public, intellectual, specific” feedback that is also relevant can lift students up.  This can reinforce the breadth of contributions that are valued.  I suppose I can imagine naming that a student has done a great job questioning how the problem worked or digging deeper into the underlying concept.  Specificity is important so that students know what is being praised.

(4) Student responsibility: creating responsibility for each other’s learning, and taking that seriously by “rating the quality of conversations groups had”, or giving “group tests” (this comes in multiple flavors).  In one version of a group test, the students work through the test together, but write it up individually, and the instructor grades only one of the individual write-ups (at random).  That will then be the grade on the test for all of the students in the group.  Another way to create inter-student responsibility is to ask a follow up question to one student in the group, and if they can’t answer it, give the group more time to talk together before returning to that same student with the question.

Justification and reasoning were also centered.  They emphasized to students the responsibility “to help someone who asked for help, but also to ask if they needed help”.

(5) High expectations: complex problems with high-level follow-up.  “Teachers would leave groups to work through their understanding rather than providing them with small structured questions that led them to the correct answer”.

(6) Effort over ability: math success is about hard work and continuing to try.  This message needs to come through.

(7) Learning practices: point out the process of what students are doing (things like fully formulating a question that they want to ask, or thinking about whether their answer is reasonable).



Relational equity: this was a learning outcome of being in the classroom, where students developed respectful relationships.


For more on Complex Instruction, see:

Cohen, E. (1994). Designing groupwork. New York: Teachers College Press.

Cohen, E., & Lotan, R. (Eds.). (1997). Working for equity in heterogeneous classrooms: Sociological the- ory in practice. New York: Teachers College Press.

Reading “High School Algebra Students Busting the Myth about Mathematical Smartness: Counterstories to the Dominant Narrative “Get It Quick and Get It Right””

Dunleavy 2018, “High School Algebra Students Busting the Myth about Mathematical Smartness: Counterstories to the Dominant Narrative “Get It Quick and Get It Right””.  Education Sciences.

I’m reading a paper about a high school Algebra I course that uses the principles of “complex instruction” (which I still need to look up).  The article highlights how students in the class, through their group process, made time for multiple solution strategies (so were not racing to complete problems), valued explanations and justification, felt “assigned competence” by their teacher (validating their contributions), and saw it as their role to help each other.

For more on complex instruction (references cited in the paper):

  1. Boaler, J.; Staples, M. Creating mathematical futures through an equitable Teaching Approach: The Case of Railside School. Teach. Coll. Record 2008, 110, 608–645.
  2. Jilk, L.M.; Erickson, S. Shifting students’ beliefs about competence by integrating mathematics strengths into tasks and participation norms. In Access and Equity: Promoting High-Quality Mathematics in Grades 6–8; Fernandes, A., Crespo, S., Civil, M., Eds.; NCTM: Reston, VA, USA, 2017; pp. 11–26
  3. Dunleavy, T.K. Delegating Mathematical Authority as a means to Strive toward Equity. JUME 2015, 8, 62–82
  4. Featherstone, H.; Crespo, S.; Jilk, L.M.; Oslund, J.A.; Parks, A.N.; Wood, M.B. Smarter Together! Collaboration nd Equity in the Elementary Math Classroom; National Council of Teachers of Mathematics: Reston, VA, USA, 2011.
  5. Horn, I.S. Strength in Numbers: Collaborative Learning in the Secondary Classroom; National Council of Teachers of Mathematics: Reston, VA, USA, 2012
  6. Cohen, E.; Lotan, R. Designing Groupwork: Strategies for the Heterogeneous Classroom, 3rd ed.; Teachers College Press: New York, NY, USA, 2014
  7. Cohen, E. Equity in heterogeneous classrooms: A challenge for teachers and sociologists. In Working for Equity in Heterogeneous Classrooms: Sociological Theory in Practice; Cohen, E.G., Lotan, R.A., Eds.; Teachers College Press: New York, NY, USA, 1997; pp. 3–14.
  8. Dunleavy, T.K. “Ms. Martin is secretly teaching us!” High school Mathematics Practices of a Teacher Striving toward Equity. Ph.D. Thesis, University of Washington, Seattle, WA, USA, 2013
  9. Horn, I. Fast kids, slow kids, lazy kids: Framing the mismatch problem in mathematics teachers’ conversations. J. Learn. Sci. 2007, 16, 37–79.

Course calendars: making a course schedule

I am thinking about how to build the schedule for my course.  Even though I’ve taught it before, scheduling is intimidating.  It has a few components:

  • There are topics that will be uncovered over the semester.  Each topic has associated informational materials (videos and text), typical student questions, follow-up check yourself questions, an in class activity, problem set question(s), skills and procedures, and assessment questions.
  • The course will meet up to 37 times, but as few as 26 times (I need to decide how many days we will meet), and the topics and learning materials need to be distributed over time.
  • The schedule of meeting days changes each semester.  This year we’ll start on a Wednesday and end on a Monday, with just one class meeting after Thanksgiving (which is a little insane – we almost always have two meetings after Thanksgiving: it looks like our semester is one MWF shorter this fall than usual).

I saw this resource from CMU, which suggests three different ways for framing out the information:…

It looks like RIT also has some design tools that could be helpful:



Reading “Learning environment and student outcomes”

I am reading “What you do is less important than how you do it: the effects of learning environment on student outcomes”, Bonem, Fedesco, and Zissimopoulos 2019 (Learning Environment Research).

They survey a large number (14,000) students across a variety of disciplines and find students ” in highly autonomy-supportive learning environments experience significant increases in satisfaction of students’ basic psychological needs, student motivation, course evaluations and academic performance”.

They reference “self-determination theory” and distinguish between courses where the focus is on pressuring students to think like the instructor (controlling), and courses where students have more ownership of the course, with their thoughts and feelings welcome.  See Ryan and Deci 2017 for information about the learning benefits of autonomy-supporting environments (environments where students “learn more conceptual knowledge, have a deeper understanding of the content, and retain information longer”).

In terms of course outcomes, autonomy, competence and relatedness are named as basic psychological needs (they say to see Deci and Ryan 1985 and 2000), and they are rated, via surveys, for each course in the study.

  • Autonomy: Options and choices can create a sense of autonomy.
  • Competence: A sense of competence is also important: this is about student perception of their progress.  Perceiving progress can lead to motivation, which can lead to improved progress.
  • Relatedness: This is about feeling like other people in the course (instructor and students) care about them and that they are contributing to the course.

The measure of learning environment for each course in the study was set by a learning climate questionnaire (LQC, 6 questions):

  1. I feel that my instructor provides me choices and options.
  2. I feel understood by my instructor.
  3. My instructor conveyed confidence in my ability to do well in the course.
  4. My instructor encouraged me to ask questions.
  5. My instructor listens to how I would like to do things.
  6. My instructor tries to understand how I see things before suggesting a new way to do things.

They also use a basic psychological needs survey (7 items are about autonomy, 6 about competence and 8 about relatedness):

  1. I feel like I can make a lot of inputs in deciding how my coursework gets done.
  2. I really like the people in this course.
  3. I do not feel very competent in this course.
  4. People in this course tell me I am good at what I do.
  5. I feel pressured in this course.
  6. I get along with people in this course.
  7. I pretty much keep to myself when in this course.
  8. I am free to express my ideas and opinions in this course.
  9. I consider the people in this course to be my friends.
  10. I have been able to learn interesting new skills in this course.
  11. When I am in this course, I have to do what I am told.
  12. Most days I feel a sense of accomplishment from this course.
  13. My feelings are taken into consideration in this course.
  14. In this course I do not get much of a chance to show how capable I am.
  15. People in this course care about me.
  16. There are not many people in this course that I am close to.
  17. I feel like I can pretty much be myself in this course.
  18. The people in this course do not seem to like me much.
  19. I often do not feel very capable in this course.
  20. There is not much opportunity for me to decide for myself how to go about my coursework.
  21. People in this course are pretty friendly towards me.

My impression is that the LQC was used to grade the learning environment of the course, and then the basic psychological needs survey was used to create dependent variables.  It doesn’t seem surprising that 1, 5, 8, 11, 13, 17, 20 of the basic needs survey (the autonomy questions) would be closely related to the LQC questions, so it seems worth focusing on the relationship between the LQC score and other outcomes of the course.

The course rating, the instructor rating, and the student rating of knowledge transfer each has a   positive relationship with the learning environment rating.

Looking at regression coefficients (see Table 3), competence seems perhaps more important than autonomy.  Competence looks to be closely related to autonomy, though, and also to a “self determination index”.  In the self determination index, students rate agreement with the following statements:

  1. Because it allows me to develop skills that are important to me.
  2. Because I would feel bad if I didn’t.
  3. Because learning all I can about academic work is really essential for me.
  4. I don’t know. I have the impression I’m wasting my time.
  5. Because acquiring all kinds of knowledge is fundamental for me.
  6. Because I feel I have to.
  7. I’m not sure anymore. I think that maybe I should quit (drop the class).
  8. Because I really enjoy it.
  9. Because it’s a sensible way to get a meaningful experience.
  10. Because I would feel guilty if I didn’t.
  11. Because it’s a practical way to acquire new knowledge.
  12. Because I really like it.
  13. Because experiencing new things is a part of who I am.
  14. Because that’s what I’m supposed to do.
  15. I don’t know.I wonder if I should continue.
  16. Because I would feel awful about myself if I didn’t.
  17. Because it’s really fun.
  18. Because that’s what I was told to do.

I suppose that this paper ends up setting up a list of 13 or so factors that are worth investigating further (6 learning environment questions and 7 autonomy questions), as well as highlighting the importance of what the authors are terming autonomy, competence and relatedness.

Notes on “How Learning Works” by Ambrose et. al., 2010

Notes based on How Learning Works : Seven Research-Based Principles for Smart Teaching by Susan A. Ambrose, Michael W. Bridges, , Michele DiPietro, , Marsha C. Lovett, Marie K. Norman, and Richard E. Mayer.  Wiley 2010.

Chapter 1: Prior knowledge.

There can be a mismatch between the prior knowledge and how it needs to be used in the new course.  In addition, “we may uncover misconceptions … [in] prior knowledge that are actively interfering with [learning]”.  Overestimating prior knowledge can lead to “a shaky foundation”.

Idea: “prior knowledge can help or hinder learning”.

Prior knowledge may be inactive, insufficient, inappropriate, or inaccurate (hinders).

Or it may be activated, sufficient, appropriate, accurate (helps).

Activation: Small prompts (“think of the second problem in relation to the first”) can help with activation.

Sufficiency:  Declarative knowledge (“knowledge of facts and concepts that can be stated or declared”) vs procedural knowledge (“knowing how and knowing when to apply various procedures, methods, theories…”).  Knowing what vs how vs when are different ways of knowing the same topic.  Identify which the students need for a new task, and assess these separately.

Appropriateness: common meanings to words may make technical meanings harder to use.

Accuracy: Isolated facts can be corrected, but flawed models (misconceptions) “are difficult to refute”.  “Conceptual change often occurs gradually” – more than a single refutation is needed.


Gauge prior knowledge: talk to colleagues, use a diagnostic (concept inventory), use self-assessment (“I have heard the term”, “I could define it”, “I could explain it”, “I could use it to solve problems”), use group brainstorming, have them create a concept map, and look for patterns of errors.

Activate accurate prior knowledge: link new material to prior knowledge from other semesters; link material to prior knowledge from the current semester; use analogies to connect to everyday experience; have them reason based on what they already know.

Address insufficient prior knowledge: identify prior knowledge requirements (distinguish what and why – declarative, from how and when – procedural.); remediate by revising the course, having 1-2 classes or review, working with a few students individually, or encouraging more prerequisite coursework.

Help students recognize inappropriate prior knowledge: when is it appropriate to apply this idea?; provide rules of thumb; be explicit about conventions and which disciplines they apply to; identify where analogies break down

Correct inaccurate knowledge: use justified reasoning problems; extra time can help students be more thoughtful; offer repeat opportunities to use the accurate knowledge.

Chapter 2: Organizing knowledge

Experts organized knowledge in meaningful ways; new learners may have facts in isolation.

Matching knowledge organization and task demands is helpful.

Knowledge “nodes” may be less connected in new learners.  In experts, information can be processed into highly connected knowledge structures.  New learners may connect topics that seem similar on the surface, while experts use underlying meaning.

Provide a structure in which to fit the new knowledge; generating organizing schemes can be part of learning.


Experts can have trouble seeing how they structure knowledge, so make a concept map to walk students through.  Think about knowledge organization in the context of a specific task.  Provide the overall organization of the course and each class meeting/week/etc.  Identify cases that illustrate looking more deeply to build connections (two things that are superficially similar or superficially different).  Connect each new concept explicitly to old ones.  Use multiple organization structures, not just one.  Incorporate concept mapping and sorting tasks.  Look for common mix-ups in student work.

Chapter 3: Motivation

Motivation is key.  Two core contributors: “the subjective value of a goal” and “the expectations for successful attainment of that goal”.

Goals: Students may be motivated by performance goals (“protecting a desired self-image” by performing a certain way in the activities).  Performance-approach goals lead to a “focus on attaining competence by meeting standards”.  Performance-avoidant goals lead to a “focus on avoiding incompetence”.  Students may have learning goals where they wan “to gain competence and truly learn what an activity or task can teach them”.  Students may have work-avoidant goals (“finish work as quickly as possible with as little effort as possible”).  Affective and social goals also have influence.  Goals may conflict, as well.

Value: Sometimes finishing a task (perhaps completing a level in a game) is satisfying.  This is attainment value.  Sometimes doing the task has intrinsic value.  A goal might be a stepping stone to some other goal, having instrumental value.  There can be multiple sources of value.

Expectancies: Outcome expectancies are beliefs about the outcomes associated with actions, such as “doing this means I’ll be able to do the problems on the exam”.  Efficacy expectancies are about whether “one is capable of identifying, organizing, initiating, and executing a course of action” to bring about the outcome.

Environment: It sits on a spectrum from supportive to unsupportive.

Three levers: value, efficacy, environment (so 8 possible combos – want a positive on all three).

low value, high efficacy –> evading behavior (this is doable but unimportant)
high value, low efficacy –> either hopeless (unsupportive environment) or fragile (supportive environment).  May feign understanding, make excuses about performance, and deny difficulty
high value, high efficacy –> defiant (in an unsupportive environment) or motivated (in a supportive environment).


Value: Connect the material to the world; provide authentic tasks; show how the material is relevant to other courses; make clear the relevance of the skills to future professional lives; identify what I value in the course and reward that int he course incentive structure; show enthusiasm for the discipline.

Expectancy: Align all the elements (objectives, assessments, and instruction).  Find the right level for challenges in the course, and create assignments at that level.  Have early assignments that provide success opportunities.  Make the course goals and expectations clear.  Provide rubrics and targeted feedback.  Use fair standards and criteria.  Share information about how attributing success or failure to internal and external factors can shape success (help them attribute success to hard work, time management, choices of study strategies, etc, and to focus on what is controllable).  Describe effective strategies.

Value and Expectancy: Offer multiple options when it is possible.  Create space and time for reflection (what did you learn, what was a valuable feature of this, what did you do to prepare, what do you need to work on, what would you do differently)




Test Anxiety and Exam Performance

I am thinking about an upcoming midterm for multivariable calculus.  I know that many students find exams stressful and feel unable to show how much they know.

I took a look at “Writing About Testing Worries Boosts Exam Performance in the Classroom.” by Ramirez and Beilock, Science. 14 Jan 2011.
One underlying idea is that anxiety or worry competes for working memory, which then diminishes performance.  My impression is that this idea is also linked to how stereotype threat exacts a performance cost.
In studies by Ramirez and Beilock, (combined: 173 students), they found that students who wrote about their thoughts and feelings before an exam, and who rated high in test anxiety, did about 5 percentage points better on the exam when they were in the expressive writing group versus a control group.
For controlled study conditions (laboratory setting), the prompt was “Please take the next 10 minutes to write as openly as possible about your thoughts and feelings regarding the math problems you are about to perform. In your writing, I want you to really let yourself go and explore your emotions and thoughts as you are getting ready to start the second set of math problems. You might relate your current thoughts to the way you have felt during other similar situations at school or in other situations in your life. Please try to be as open as possible as you write about your thoughts at this time. Remember, there will be no identifying information on your essay. None of the experimenters, including me, can link your writing to you. Please start writing.”

In actual exam conditions (ninth grade biology final exam), the prompt was “We would like YOU to take the next 10 minutes to write as openly as possible about your thoughts and feelings regarding the exam you are about to take. In your writing, I want you to really let yourself go and explore your emotions and thoughts as you are getting ready to start the exam. You might relate your current thoughts to the way you have felt during other similar situations at school or in other situations in your life. Please try to be as open as possible as you write about your thoughts at this time.  There will be no identifying information on your essay. None of the teachers can link your writing or any other information to you. If you finish early, please just sit quietly and wait for the teachers’ instructions. You may end up sitting quietly for several minutes while your classmates finish the tasks they were asked to do.  That’s ok. You will be given plenty of time to complete the upcoming exam. This task will only take about 10 minutes in total. Please begin”

Webtools for math teaching

This semester I’m using Gradescope for homework and exam grading.  It is working relatively well, although not perfectly.  This enables us to reuse feedback, makes it easy for me to review regrade requests, and gives us access to all student work in the course.

For our discussion board, we’re using Piazza, mainly because it’s easy to type in the math via their built in Latex.

Online homework, via Webwork,  forms a portion of each homework assignment.  These are mainly exercise type questions so that students can receive immediate feedback on whether they are applying procedures properly.

I am curious to take a look at iMathAS at some point.  Webwork has a nice bank of questions from the text our course is aligned with, and there is some convenience to that, but it seems worth checking out other options.