This are notes from a presentation and conversation with the CSU Faculty Development Council. The general theme is how to work with mathematics instructors, spelling out things that might not be obvious to outsiders. -* Eric Hsu, November 7th, 2017*

**My Background******teaching college math since 1989, tenure track since 2001- I've been supporting almost every kind of person teaching math for years.
- grad instructors (& some lecturers)
- worked as instructor supervisor for Treisman workshops at UC Berkeley 1994-1998.
- graduate instructor supervisor at SF State 2001-present, 3-unit workshop
- developmental algebra co-director, 2005-present

- tenured faculty
- workshops for MAA, 2006-2008 (Emerging Scholars workshops), Transforming Calculus 2007-2009, Google Faculty - Math of Khan 2011-12, Hybrid Precalculus 2013-2015, Capstone Faculty Resources 2015-2016.

- non-math STEM faculty
- (NSF INCLUDES, Math Gateway, Foundational Science, Engineering Pathways, 2010-present)

- also secondary and elementary teacher PD continuously from 2003-2017, pre-service and in-service

**Reform Math**- As an example, a five minute lesson (Flag Hoist)
- Engagement, discourse, critical thinking, differentiation, strategic metacognition
**Question for you: Other pedagogical ideas to set in a math context?**- I've taught algebra, calculus, capstone level math this way. People have taught upper-division math courses with team problem solving. It can be done with the right instructors with the right support to change and fail/iterate.

**The Math Wars, Then**- In the 1990s there was a Reform Math movement and a vigorous reactionary movement. This created the Math Wars, pitting Reform and Conservative sides against each other. The battleground was usually a district or college department adoption of a reform curriculum. The conservatives (led by research mathematicians, e.g. at Stanford, UCB and CSU Los Angeles) would point out math errors and claim the math being taught was wrong/unimportant, and the reformers were incompetent/evil. Very bitter and personal battles.
- Conservatives allied with parents wanting their children to keep elite education tracks (e.g. calculus in G11/G12 for elite colleges) by preserving status quo.
- Many of the faculty directly involved are older and about to retire. So when I joined SFSU in 2001 I was asked which side of reform I was on.

**The Math Wars, Now**- Younger faculty have experienced only the loud echoes of the Math Wars. Many experienced group work or reform math of varying quality, and so the identity politics of Math Wars have mellowed. But faculty 45 and older probably had to pick a side, and the default was Conservative.
- Reform no longer considered radical break, but a debateable choice.
- Detente with development of the Common Core Math Standards. Reform people like McCallum worked with H. Wu. Mathematical Practices highlighted, not just laundry list of topics.
- (victim of war against high-stakes testing and national standards)

- The MAA (the teaching-heavy math professional organization) is fully behind reform math.
- The AMS (research-heavy math professional org) is drifting reform-wards, but still mostly neutral.

**Cultural Upbringing**Research math talks are culturally required to be incomprehensible.

Audience must not be able to follow the talk by halfway. A good speaker can let people follow up to half of it, but no more. Otherwise your work is too easy and obvious and you are stupid.

Math PhDs almost all teach as grad students.

A little knowledge is a dangerous thing. At most Research I places, neutral-to-negative encouragement for teaching. Students obviously a burden, see how the faculty role models treat them.

Most of them have been students in math classes where half the students failed. That's not considered shockingly poor teaching, it's considered a tough standard, and they are proud for surviving.

**Common Math Instructor Folk Beliefs**- Many believe the following because they were cultural assumptions held by their own teachers.
- Teaching means getting students to the next rung on the ladder.
- Thus, can't understand how one could use algebraic thinking at a college level.
- Problem solving, logical argument, representations. Math for powerful argument/analysis, beyond rote recipies.
- Apply material to novel settings (science - decibels and pH, fanciful settings - magic pizza oven, advanced math - linear programming)
- Also don't get that students have already learned developmental material once, twice or more. Need more than just The Same, But Louder.

- I transmit clearly, students receive clearly, imitate, practice, ask questions.
- No evidence of this. Student brains have plenty of ideas that need to be repacked/built upon.

- Students who don't get it weren't the chosen ones ("math people") or won't work hard enough ("lazy kids", "unmotivated"). If a lot of my students fail, that's because I have high standards.
- No, everyone can be a "math person". And how come more students pass and remain if you teach more engagingly?

- I'm welcoming, since I have office hours. Students who don't come talk to me just aren't motivated.
- Students should be motivated only by grades and love of math. Everything else is superficial entertainment.
- No, they love to be right, to impress each other, to understand, to pursue curiousity, and to be able to do new things.

- You can't learn calculus without mastering pre-calculus. No pre-calculus without mastering algebra. No algebra without mastering fractions. No fractions without mastering quick arithmetic.
- (These are all obviously false because 95% of learners in those classes haven't mastered it. You don't really master it until you teach it, more than once. So spiraling is unavoidable.)

- You can't learn math without rigorously deducing it from axioms.
- No, the level of logical rigor demanded by math teachers is mostly not demanded by science client disciplines.

- I'm the teacher because I can do math faster and more precisely than you.
- Thus it's VERY threatening to do math in front of other math teachers, because a mistake destroys your superiority and exposes imposters.

- Things were better in the old days when we had rigor and standards and students knew their stuff. Now it's watered down.
- The usual Good Old Days dream. In fact, previous generations did worse on algebra/math tests.

- Many can't see how to make material more accessible.
- Blurry/no distinction between understanding material and being able to perform mechanical testable algorithms on tests.

**Stages(!) of Instructor's Teaching Theories**- what teachers should do (How should I act?)
- e.g. write clearly, wait time for answers, friendly complete syllabus
- "What do I do if students won't break into groups?" "What do I do if a student cries?"

- how students mess up what teachers ask (How do others respond?)
- e.g. "students always mess up fraction division", "student have a tough time with Y", "so many students surf the internet", "students are addicted to calculators"

- how students are thinking about math (How do others think?)
- e.g. "these students are seeing these symbols as strings to manipulate and need help to give meaning to them"

- how students grow in a class (and out-of-class) environment (How do others grow?)
- e.g. "this task will give students who are strong at creative interpretations a chance to shine, but also let's pull in students who graph it", "This problem rewards exploration and not just solving it quickly with symbols. That will give status to students who aren't used to being faster at math tasks".

- like developmental stages of intellectual empathy

- what teachers should do (How should I act?)
**Stages of Trust in Faculty Collaboration**- share materials (just don't do the math together)
- visit each other's classroom
- try non-trivial math together
- discuss curriculum tasks in detail by actually trying problems to see affordances and pitfalls

**Defense Mechanisms & The Team**- "This isn't real/important/rigorous math." Need someone buying in with at least the highest pure math terminal degree of the group (PhD for tenure-track faculty, MA for lecturers).
- "This won't work for my students." Need someone who's taught the exact student body
- "Abstract education theory doesn't make sense in a math class." Need someone who's thought hard about how pedagogical ideas play out in a math classroom. E.g. equity of voice, more than just letting students ask questions. How do we grow people's voice? Help them grow their mathematical precision and deductive logic?

**Math Faculty as Advocates**- Most math teachers are survivors of a painful education process and need compassion and safety.
- Often curmudgeonly, but can often be swayed by evidence (student results), and by having their minds expanded as to what is evidence and what is possible (examples of engaged students learning interesting math).
- Math education is really cognitive theory an hence will never be as rigorous as math. You'll never prove a result. But you can get observational evidence and there are also analysis surveys and tests. And you can start attending to student morale/attitude/coorperation/identity/concept of mathematics/communication and argumentation/creativity without abandoning deductive logic.

- Every math teacher knows in their gut the difference between being able to do math and understanding it. But for many, they need mathematical help seeing there is more to understand in the elementary math than they themselves learned, since they were so swift at calculating. Threatening, but very liberating when they see it.
- Most math teachers would like to have positive classrooms and all the good stuff, if they can be convinced that students don't lose math skills in the change.
- Once sold, they are surprisingly good advocates. Partly because they're known to be the most conservative group, so if they're going along with reform, it must be okay.
- Also, science faculty are intimidated by math faculty (science faculty have relative status in proportion to the amount of math their field uses), so reforms in math might be used to support reforms in science teaching.

**Lecturers**- often freeway flyers, different institutions have different expectations and cultures and tolerance for reform. Go for least common denominator of lecture.
- can't afford to get bad student evaluations w/o security
- care greatly about teaching
- highly underpaid
- lower class citizen of departments

**Resources**