1. Teaching for Understanding: What Will It Take?
Today: talk about teaching – the final common pathway!
Not the only thing, but the only thing we, as instructors, have control over.
In particular I want to talk about teaching for understanding.
We want our students to:
See structure
Make analogies
Solve novel problems
Flexibly bring knowledge to bear!
My own background is cross-cultural: looking at math and science teaching around the world.
What have we learned from this?
Today:
Start with students: cc students’ views of math
What it takes to teach for understanding: mine and others’ research
How can we do this in our classrooms?
Jim Stigler
Honolulu
March 5, 2016

2. What do community college developmental mathematics students know about mathematics?
We know lots about these students, but what do they know about math?
To me, interesting… it’s the result after K-12 math education.

3. Developmental Math Students: Views of Math
View math as rules/procedures to be memorized; can’t “figure it out”
When students were asked, “What does it mean to be good at math?” 77% gave answers like these:
“Math is just all these steps.”
“In math, sometimes you have to just accept that that’s the way it is and there’s no reason behind it.”
“I don't think [being good at math] has anything to do with reasoning. It's all memorization.”
You can see this when you ask students a question that requires thinking. They are trying to remember; to see what the professor wants; but they don’t seem to think they can think through a solution.
Developmental Math Students: Views of Math

4. Developmental Math Students5 8
Which is greater: or ?
53% correct, but very few could explain why
Compulsion to calculate…
5a = a8
5 = 8 ✔
Developmental Math Students

5. Developmental Math Students
462
+ 253
715
Interviewer: “How could you check that your answer is correct?”
715 – 253 = 462
Interviewer: “Why did you subtract 253? Could you have subtracted 462 to check the answer?”
What disturbs me is not that these students can’t pass the test, but it’s the lack of understanding.
They knew how to answer this when they were 5 years old!
How did this happen?
Developmental Math Students

6. “Every system is perfectly designed to achieve the results it gets.”
-Paul Bataldan
Teaching is a system.
First step in improving a system is to see the system the way it is now.
How did our students come to have these views? Clearly they are the product of our K-12 education system.
But how do these views emerge?
Teaching, it turns out, is not an easy system to study because it is cultural…
Learned implicitly
Hard to see
Hard to change – multiply determined

7. TIMSS Video Studies: How We Teach Mathematics
Because teaching is cultural, comparing across cultures helps us see our own system more clearly.
E.g., confusion in Japan.
And it turns out, teaching varies a lot more across cultures than within.
TIMSS Video: video survey designed to study average teaching.
TIMSS Video Studies: How We Teach Mathematics

8. Math Teaching in Japan vs. US
Japan is one of the highest achieving countries in mathematics.
How do they teach compared to us?
Goals: skill vs. understanding
Lesson: teach how, then let students practice vs. struggle with problem, then connect with concepts (discussion or lecture)
Math Teaching in Japan vs. US

9. Japan: “Slow and sticky”
US: “Quick and snappy”
Japan: “Slow and sticky”
Hess & Azuma:
Quick & Snappy vs. Slow and Sticky
They also argue: Every society solves the problem of motivation in a different way.
Also: US teachers think teaching math is easy; Japanese teachers think it’s hard!
SOURCE: Hess & Azuma (1991)

10. Productive struggle – with important mathematics
Explicit connections – among concepts, procedures, problems, situations
Deliberate practice – increasing variation and complexity over time
These things differentiate Japanese math instruction from our own.
They also are supported by a large body of cognitive science research on learning.
And, although other high achieving countries teach differently from each other, they all manage to create these kinds of learning opportunities for students.
Research Indicates Three Types of Learning Opportunities Required for Deep Understanding

11. Productive struggle –
– desirable difficulties

12. “I never enlighten anyone who has not been driven to distraction by trying to understand a difficulty or who has not got into a frenzy trying to put his ideas into words. When I have pointed out one corner of a square to anyone and he does not come back with the other three, I will not point it out to him a second time” (不愤不启, 不悱不发. 举一隅不以三隅反, 则不复也.) (Lau, 7:8)
Question: how do we get students to struggle?
Important: relationship/love/respect for teacher
Confucius

13. The Power of Connections
Few Connections
Many Connections B B C C A A D D F F E E G G H H
The Power of Connections

14. Example: Struggle to Make Connections
What fraction of the rectangle is shaded brown?
(a)
(b) ⅓ ⅓
This illustrates all three learning opportunities: the increasing variation/complexity (deliberate practice); the struggle (hard for students); the connection to explicit concepts (equal size pieces, keeping track of what the “whole” is)
Example of equivalent equations: need to keep students connected to the core idea.
Another example: |5x| = 10 vs. |5x| = -10
Example: Struggle to Make Connections
SOURCE: Deborah Ball (2013)

15. Deliberate Practice Different from repetitive practice
Constantly increasing variation, complexity, challenge
With feedback
Staving off premature automaticity
Understanding takes practice, too!
We think of understanding as sudden, but this isn’t really the case. It also requires practice, of a certain kind, over time.
Deliberate Practice

16. How can you create these learning opportunities for your students?
First of all, you don’t need to copy Japan.
All the high achieving countries in our study differed in their teaching from each other.
But all found some way, in their situation, to create these learning opportunities.
Problem is, teaching is highly contextual. If we ask a Japanese student to work on a problem they’ve never seen, they will struggle. But if we ask American students, they may just give up. In different cultures will require different methods.

17. Shooting a rocket to the moon, or driving to work?
Teaching is more like driving to work than shooting a rocket ship to the moon.
Only you can figure out what will work for your students.
Shooting a rocket to the moon, or driving to work?

18. Sorry for the long , I am just one of those people that when it comes to math, I need step by step instructions in order to retain what I am learning, I don't know about others, but I get extremely confused if the directions are not laid out clearly. I'm not sure if there is anything you can do about future assignments, but I just wanted to bring it up, instead of not saying anything at all and letting myself get frustrated, like I've noticed some other people have in the class.
And of course, no one likes change; students will push back.
from a student