Why (and how) I teach without long lectures – J



Why (and how) I teach without long lectures : J. J Uhl

Why I gave up long lectures

1. Motivating Questions:

o How would you IDEAL teacher teach?

o What would the students look like during class time?

2. MAIN IDEA: Technology can replace lectures if used correctly. To engage in mathematics you have to play around with ideas.

3. “I have completely abandoned the long lecture method.” - Uhl, University of Illinois Professor of Mathematics

4. Lectures did not produce the results he liked, students couldn’t perform as he had hoped.

5. “… students do not get much out of long lectures, no matter how well they are constructed.”

6. Students became “quiet scribes”

7. Students demand ACTION and ACCOUNTABILITY.

8. Warning flags:

o “Why won’t my students talk to me?”

o “Why is class attendance so poor?”

o “Why won’t students do their homework?”

o “Why do they perform so poorly on exams?”

9. The answers teacher come up usually are centered around students faults.

10. Lectures are ineffective for introducing new material. “They are full of words that have not yet taken on meaning and full of answers to questions not yet asked by the students.”

➢ What’s the difference between listening to a lecture and reading lecture notes? Why go to class when you have I-Notes?

➢ Use sound bites and short follow-up lectures.

➢ It sounds like it’s a question of timing. When is a good time for a lecture style of instruction?

11. Professors’ tendencies are to think for the students.

➢ What does this look like? How/is this manifested in class? During tutoring? In teaching?

12. The educational levels between the professor and the student are very different.

o Professor says things that SHOULD be known.

o Students struggle with WHAT they know and understanding at a BASIC level what they SHOULD know.

o This results in easier tests/quizzes/HW.

13. Books are quick to show examples of HW problems.

What I replaced lectures with

1. “generalizations are best made by abstraction from experience.” – learning kitty.

➢ Uhl claims that one of the benefits of having technology based classes is the ability to produce limitless number of examples. What role in the education of a student does having access to any number of examples play?

➢ Picture.

2. Teachers take a back seat to student discovery. (e.g. Students come with questions night before weekly HW is due.)

o “No longer are students the professor’s audience; students are the professor’s apprentices.”

➢ What kind of questions are being raised the NIGHT before the assignment is due: “How do you do problem 4?”

Content Issues

1. Math content for the most part has not changed, in part to the lecture system.

2. Math texts have also become “one-size-fits-all” texts (e.g. The big grey book: Math 120, Math 130, Math 242)

3. In response to this, Applied Math and Engineering classes have offered their own crash course in necessary math content areas.

Specific Remarks about Steven Krantz’s Revision

1. The goal is to get students to think ANALYTICALLY and CRITICALLY.

2. “Have the students to play with graphics, first varying a (of ax^2+bx+c) and coming up with a conjecture of what the influence of a is. Then ask them to explain how changing the value of a affects the first and second derivatives in a certain way and how this is reflected in corresponding plots.”

➢ You have to have the right activities followed up with the right questions. How can you ensure that students are learning something and not just goofing around?

3. Process of Mathematics learning: Intuition – Trial – Error – Speculation – Conjecture – Proof

➢ Is this feasible? Can we learn in such a developed way for every math concept? Depth or scope?

4. Claim: students are capable of asking thoughtful questions in front of a computer screen.

➢ Math as a laboratory science … Open-ended labs vs. lecture guided labs … How much control should teachers give up? Are they willing?

Let’s Abolish Pencil-and-Paper Arithmetic : Anthony Ralston

1. Motivating Questions:

a. What are the most important skills to succeed in mathematics at any age?

b. Where does computation lie?

c. What are initial reactions to calculator usage at early grades?

d. List some differences between solving 2*3(-4+3)^2 with paper and with a calculator.

2. MAIN IDEA: Curriculum focused on mental arithmetic joined with increased calculator use can lead to stronger mathematics students.

3. Abolish Pencil-and-Paper arithmetic (PPA) exercises: basic multiply/divide/add/subtract problems.

o Kids should not have to be taught this nor tested on it.

However,

4. California has adopted a PPA-only regimen

5. British Secretary of State for Education, David Blunkett, discourages, “as far as possible the use of calculators.”

6. Computers – good … Calculators – Bad

➢ What is it about computers that place it on a more desirable scale than computers

7. Research mathematicians – proponents of pulling the plug on calculators.

8. Reform is being made from outsiders looking in. They don’t really understand elementary schools.

9. “They have diagnosed the problem correctly – poor preparation for university mathematics – but they don’t understand the causes of the problem or its cure.”

➢ Why is research and educational practice disjoint? Why not have every teacher as a sort-of “field researcher”. How come educational research has such a strong say in educational practice?

The Value of Learning PPA

1. Two factors in the anti-calculator movement:

o Poor mathematics compared internationally – Belief that failure at proficiency in PPA is a stumbling block in learning higher math.

o Progressively poorer preparation for first-year university students to study university mathematics – Belief that calculators contribute to the lack of rigor in learning math, a dumbing down of the curriculum.

However,

2. With respect to PPA : “ … the message of why one does arithmetic is lost in the emphasis on computational accuracy.”

o PPA - You get questions like, “Is this the right answer?” (erase, erase) “How about this.”

3. It should be one side or the other, 100% PPA or 100% calculator.

4. Classical PPA instruction has lost its use in a world where most arithmetic is done using the calculator.

➢ What’s more important: Knowing WHAT to do or knowing HOW to do it?

➢ Is it possible to know HOW to do it without WHAT you’re doing? Black box?

5. “PPA stands or falls insofar as this skill is necessary to learn subsequent mathematics.”

6. Unfounded Beliefs:

o Calculator = pushing buttons : No research supporting this?

o Calculator use in elementary grades = less work, dumbing down curriculum and loss of time spent in other topics of primary school mathematics.

7. It is not the KNOWING HOW but the “ancillary benefits” of computation: Numeeracy, number sense, knowing strategies to check arithmetic.

8. What makes having a class based on calculator use for computation is the loss of “technique” – the ability to understand and manipulate symbols.

➢ Can calculators fully incorporate technique? Can students understand the workings of the “Black Box?”

9. Does PPA have some inherent abstraction that calculator use doesn’t?

10. According to “his research” there is no evidence that supports the claim that a calculator “impedes children’s understanding of arithmetic or acquisition of later mathematics.”

Mental Arithmetic

1. Why? It’s more efficient. More so than calculator or PPA

2. It is possible to learn PPA by rote and thus lose out on all “technique” or abstraction.

➢ Perhaps it seems more abstract because you’re writing out steps? There’s some visual proof of what the type of thinking that SHOULD be going on?

➢ Who says we should learn to do PPA all the same way? There’s more than one way to multiply. (e.g. figuring out tips or tax)

3. Mental arithmetic can be rote as well.. Simply memorizing multiplication tables.

4. Allow kids to explore different ways of multi-digit mental arithmetic. Let them figure out how they understand things individually.

5. “Multi-digit mental arithmetic ability requires just the kind of mind developing in logical thinking that mathematicians have always believed is an advantage of studying their discipline quite aside from whatever subject matter is learned.”

6. Mental arithmetic = breakdown in importance of algorithms?

7. Long division – time it takes to teach outweighs the benefits.

➢ Are there easier ways to learn about division? Remainders? What is it?

8. Mental arithmetic can benefit from calculators: “the importance of mental calculation inevitably has implications for a judicious use of calculators.”

o “Calculator users must be able to estimate mentally the results of their calculations.”

Elementary School Mathematics without PPA

Author’s idea:

1. Stress mental arithmetic as soon as any arithmetic is taught.

2. Calculator usage should be encouraged.

o Exams should be calculator friendly, except where mental arithmetic is assessed.

3. Manipulatives and other mathematical models are allowed.

4. Now that you’re not doing so much drill-and-practice PPA, you have time for other math topics.

➢ Why do we seemingly spend 5 or 6 years PRACTICING computation, if there is technology that can do it for us and quicker?

Secondary School and Later Mathematics

1. Mental algebra in secondary grades is the analog of mental arithmetic in elementary school.

2. The aim of secondary and higher mathematics seems to be to create the “student-machine”. But students still only have a mechanical grasp of the material.

3. This would be great if elementary teachers had better math understanding. Most of the time they’re trained in PPA, and not interested in mathematics.

o Solution: Specialist math teachers?

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