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Dan’s Reading Notes on Krantz

For your consideration; Dan’s reading notes on Krantz’s How to Teach Mathematics (PDF):



Student Expectations; CUNY Remediation Changes; Biology Supplements


  • Steven G. Krantz, How to Teach Mathematics, Chapters 4-6


  • Daniel, Kristin, Tian


  • Students being surprised at learning any new skill in a college math class (after many years of algebra review). Complaints/grievances.
  • Students being accustomed to scaling/extra credit to make sure they pass (instead of rigorous standards & hard work). History of K-12, etc.
  • Major CUNY changes to placement, remediation, low-level courses, and high-level prerequisite accounting (i.e., eliminate high-level courses in STEM majors?).
  • Likely very few M2 courses in future (only for STEM students who do not pass elementary algebra placement). New non-algebra course for non-STEM majors (majority of clientele). For M2, CUNY uniform final exam will not require passing grade (same for English writing test).
  • Begs question: Will placement tests still be given? Will uniform final still be given, or eliminated?
  • Biology department offers 1-2 credit math supplements linked to sections of non-major biology (e.g., BIO 33). Custom workbook.
  • Students pt-in, scattered through other sections.
  • Majors are given a 4-day workshop in math before the semester begins (grant-funded; supplies food, mentors, core textbook at end).
  • Pre- and post-surveys.
  • Referring students with prerequisite deficiencies to consultations in office hours, and/or the math workshop.
  • Are all freshmen now in learning communities?
  • Can we get a supplement section to prerequisite skills for math courses?


Tracking Math; Outreach Ideas; Exercise Ideas; Takeaways


  • Steven G. Krantz, How to Teach Mathematics, Chapters 3-4


  • Daniel, Patrick, Kristen, Aleksandr, Deborah, Eva, Tian (and Sara K.)


  • What else can this group do to move forward with better math teaching at KCC?
  • STEM Vision Committee — team up with them
  • Run a pilot somehow using new strategies to teach math, and then follow students to see how they do in chemistry, biology, etc.
  • With the doing away of math requirements (algebra) for non-STEM majors, we are separating STEM and non-STEM tracks early and with little chance for switching tracks later in education or career without starting over from square one.
  • Elementary Algebra M2 is now a multiple-choice test, with a calculator, and students do not have to pass the test to pass the class. As of next semester, students do not need to even pass this class to graduate.
  • Given all of these things we can’t control, is there anything we can do? Would expanding the Math Workshop help? It would help some students, but probably not a majority.
  • Some research claims remediation never solves the problem — ongoing support will always be needed.
  • What other things can we do? Give more practice — students do know some math, but need practice & feedback.
  • Physical sciences uses ALEKS (others: MathXL, WebWork.)
  • One observation: Even with ALEKS homework 25% of grade, some students still don’t do it. Others find online homework helpful.
  • Another strategy is to spend some time on definitions, so that the words in word problems are understood.
  • Ask students to draw what the instructions say (show them an example; in chemistry, physics, biology).
  • CHE 12 “all about logs”, but only prerequisite is MAT 9; logs covered briefly in MAT 14.
  • Many students are trained to be passive learners, so much so that many consider preparing for class & doing the exercises ahead of time on their own as cheating.
  • How has this FIG affected your teaching practice? For most, understanding what math is needed for science courses, vs. what math is taught in prerequisites, has been revelatory and helpful.


K-12 Preparation; Cultural Study Habits; General Principles at Start or End?


  • Steven G. Krantz, How to Teach Mathematics, Chapters 1-3


  • Aleksandr Gorbenko, Patrick Lloyd, Daniel Collins, Kristin Polizotto, Tian Cai, Deborah Berhanu


  • Question: What did you agree or disagree with?
  • Moore Method: Surely inappropriate for our students.
  • Question: Why are KCC students unprepared?
  • K-12 prep as a problem for KCC students.
  • Math anxiety among K-6 teachers, esp. females, who then communicate this to students (esp. girls who identify with teacher).
  • The students get high school diploma without really learning math.
  • Comparison to teacher qualifications in different countries/states.
  • Specialist K-6 teachers would be the #1 thing to change (DRC).
  • Working in groups outside of class — effective but possibly a challenge for our students — culturally, time constraints, socially, etc.
  • Groups only effective if carefully chosen — someone who knows what they’re doing.
  • Asian students may know how to do the problems & get good grades, but necessarily able to apply or engage with the world with that knowledge.
  • Influence of home environment (in terms of academic discipline) and of social class.
  • Do the suggestions in this book help all students? Effective across the board? (Probably.)
  • Students cannot set up a problem — logically — because that unit is skipped in K-12.
  • Logic not required at KCC (even in math department) — inductive & deductive reasoning is not required.
  • Krantz recommends examples, then building/developing theorem.
  • Other option: Principle/theorem, then proof, then examples (traditional in definition-theorem-proof-application).
  • Daniel thinks traditional is better for students who will not be mathematicians. But Tian/Pat thought examples earlier on was better.
  • Next meeting: Nov-22nd at 12:40 PM (to include outside grant observer).


Prerequisite Challenges; Testing and Time; K-3 Experience


  • Steven G. Krantz, How to Teach Mathematics, Preface


  • Daniel Collins, Emral Devany


(1) Prerequisite Challenges

  • Started with an overview of the book, as some people were just being informed/picking it up today. In future sessions we’ll plan to cover the main chapters, and try to find ways to attract more attendees.
  • Discussed challenges in classes where many students don’t have the expected prerequisite math skills. (For example: Integrated Seminar in Biology, including majors and practicing nurses).
  • Examples: Estimated 1/3 may not be able to compute a percent increase between two numbers (or its inverse). This would be used in a metabolism problem, etc. Note that this was previously tested on the CUNY-wide remedial CEAFE (CUNY Elematary Algebra Final Exam), but was recently removed as it was considered too difficult.
  • Unfamiliarity with efficient conversion methods: Biology, using the metric units (multiplying by 1000 vs. moving point 3 places). Statistics, converting percent to decimal (dividing by 100 vs moving point 2 places).
  • Problems with students knowing algebra but not arithmetic. (CUNY entry testing with COMPASS aborted testing on arithmetic if algebra was passed; this may be reversed with new Accu-Placer test, which combines both in a single test/score).
  • Students passing math via raw, herculean memorization of procedures (much harder). Understanding concepts can make these methods trivial to remember.
  • Problem of instructor not realizing basic things that students don’t know, possibly for many semesters or years (e.g., types of energy, conversions, checking solutions to equations).
  • History of repeating definitions without quantitative applications in some prior science classes.

(2) Testing and Time

  • Test-taking history, where students are mostly exposed to multiple-choice tests and tricks to pass them (Krantz in Ch. 2 strongly recommends non-multiple-choice tests for this reason).
  • Need to test/quiz on exactly the basic skills practiced in class (not more exotic extensions).
  • But: Time constraints on grading, plus assignments, may bias towards multiple-choice tests.
  • Suggest: Tests that are part multiple-choice, part short answer (possibly fewer in larger classes).
  • Suggestion: Using a point-and-click Rubric in Blackboard for quickly grading assignments and providing feedback (e.g. in a programming course).
  • Krantz in Ch. 1: Time management in class: How to avoid an activity that goes long, cut off in middle by end of session, need to pick up in middle next time?
  • Prepare 2-3 short exercises in class so one or more can be cut for time if needed.
  • Permitting students to go early if they complete exercise quickly: Can give extra attention for other students (with reduced anxiety over struggles in front of peers).

(3) K-3 Experience

  • Current student experience in K-6: 1st grade — Test correct performance on several defined procedures. 2nd grade — Give credit for any generation of correct answer, no matter how. Instruction on “how to take multiple-choice test” tricks. 3rd grade — Standardized multiple-choice tests begin, used to assess schools. Seemingly large strategic shift at the point when school needs high grades on standardized tests.
  • Common Core complaints: Often unjustified.
  • College students not knowing 2nd-3rd grade concepts (scientific method, etc.)
  • Passing all students K-12 (no hold backs).


Fall 2016 Info: How to Teach Mathematics

Fall 2016 Book: Stephen G. Krantz, How to Teach Mathematics (3rd Ed.), published by the AMS (American Mathematical Society), 2015. This an update from earlier editions published in 1993 and 1999.

Book Availability: KCTL has a supply of books for our fall reading. If you can commit to joining at least two of our fall discussion meetings, then you can pick up a book for free in M-391.

Discussion Meetings: Mondays, 11:30 AM – 12:30 PM, in M-391 on the following dates.

  • Sep-26: Ch. 1-2, “Guiding principles”, “Practical matters”.
  • Oct-24: Ch. 3-4, “Spiritual matters”, “The electronic world”.
  • Nov-21: Ch. 5-6, “Difficult matters”, “A new beginning”.

More Optional Readings: Available online.

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