 There has been a consistent movement toward more Application and less Complexity.  Increased number of questions asking for the why and what does it all mean?  Heightened need to combine several learned concepts into one question.  Stronger need for students to be able to problem solve without a particular algorithm.

Now, let’s just answer c)

Extend this beyond just increasing. Why? How would concavity effect a midpoint approximation?

 Other than a short review of functions and trig, start the course off with immediate challenges.  Find something they think they know and challenge them to explain it. Ex:  Or how do differentials relate to local linear approximation?

 As each new topic is developed, make a conscious effort to see how it relates to earlier sections.  How are the Intermediate Value Theorem, Mean Value Theorem and Average Value for Integrals related?  As the course progresses, the teacher/professor takes a more limited role.  Final few weeks, the instructor should sit within the classroom posing as a curious student.

 Mathematical Practices 1: Make sense of problems and persevere in solving them 2: Reason abstractly and quantitatively 3: Construct viable arguments and critique the reasoning of others

LEVELED Math Questions CHECK FOR ACCURACY You KNOW the student understands 1.Would that work if you didn’t use that method? 2.Can you create and solve a problem similar to this one? 3.Can you make a model to show that? 4.Can you use a different method to show your thinking?

LEVELED Math Questions CHECK FOR UNDERSTANDING You THINK the student may be confused 1.What do you need to find out? 2.How would you describe this problem in your own words? 3.What pieces of this problem make sense and which pieces are you confused by? 4.Could you try this with simpler numbers? 5.Have you tried pictures or manipulatives?

LEVELED Math Questions GIVING CLARITY You KNOW the student is confused 1.Which words are important? 2.Where do you think we should start? 3.What is the goal of this problem? 4.Can you explain the steps you think we should take? 5.How can your group members help you? Draw it? Talking it out?

 “The humble question is an indispensable tool, the spade that helps us dig for truth, or the flashlight that illuminates surrounding darkness. Questioning helps us learn, explore the unknown, and adapt to change.”  Warren Berger (Edutopia) The Power of Questions

 Make it Safe  Make it Cool  Make it Fun  Make it Rewarding  Make it Stick

Why 5 + ? Don’t they just have to be zeros of the second derivative to be points of inflection? Could you show this graphically?

What is different about this compared to the last one?

How did we go from here to here?

 Success comes from good choices.  Good choices come from experience.  Experience comes from Making Mistakes.

Brain Research

 6. Attend to precision  8. Look for and express regularity in repeated reasoning Mathematical Practices

 ATTEND TO PRECISION  Communication  Presentation  Mathematical Notation

Find the Derivative Could you find a pattern with any linear over linear to a power?

Shulman’s Lemma

 4. Model with mathematics  7. Look for and make use of structure Mathematical Practices

How could a drawing help here? Most students learn Riemann sums with regular subintervals. Ask for further explanation here.

Mathematical Practice #5 Use Appropriate Tools Strategically