Encouraging Perseverance in Problem Solving: Strategies and Resources
Robert Lochel
Hatboro-Horsham School District
R. Lochel. ATMOPAV 2012
FROM THE COMMON CORE “STANDARDS FOR MATHEMATICAL PRACTICE” STANDARD 1: Make sense of problems and
persevere in solving them. Mathematically proficient students start by
explaining to themselves the meaning of a problem and looking for entry points to its solution.
They monitor and evaluate their progress and change course if necessary.
Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
R. Lochel. ATMOPAV 2012
DO THESE PROBLEMS ENCOURAGE PERSISTANCE? Landscapers plan to spread a layer of stone on a path. The
number s of bags of stone needed depends on the depth d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2 inches deep. Write a direct variation equation that relates d and s. Then find the number of bags needed to spread a layer that is 3 inches deep. From Larson Algebra 1
Main Street and Market Street intersect to form an 80-degree angle. Use a protractor and a ruler to sketch the intersection.
From algebra.com: “One good use for rational equations is the shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?”
R. Lochel. ATMOPAV 2012
STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION Landscapers plan to spread a layer of stone
on a path. The number s of bags of stone needed depends on the depth d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2 inches deep. Write a direct variation equation that relates d and s. Then find the number of bags needed to spread a layer that is 3 inches deep. Direct Variation, Linear Functions – Algebra 1 Table Tennis Problem: see links below
R. Lochel. ATMOPAV 2012
STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION Main Street and Market Street intersect to
form an 80-degree angle. Use a protractor and a ruler to sketch the intersection.
6th Grade – Angle Measurement, Acute, Obtuse angles.
“Sharp Turn” problem
R. Lochel. ATMOPAV 2012
STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION “One good use for rational equations is the
shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?”
Rational Functions and Equations A.K.A.: the most fun unit of the year!
“Looking for an Assistant” video. Link below.
R. Lochel. ATMOPAV 2012
STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS Conic Sections Drawing Project – Algebra 2
My! How the technology has changed! DOS drawing program – black, white AND
magenta! Math Toolkit grapher – use Print Screen! NOW: www.desmos.com
Lecture on Conics
Test (bombed it!)
Drawing Project (if we fit it in after all the snow
days…)
Introduce
Drawing Project
Let’s explore
our conics!
Do we need
such a big test?
R. Lochel. ATMOPAV 2012
STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS
R. Lochel. ATMOPAV 2012
STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS
R. Lochel. ATMOPAV 2012
STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS
R. Lochel. ATMOPAV 2012
STRATEGY 3: EMBRACE ALTERNATE MEANS OF “TURNING IN” PROBLEMS “Elvis on the Beach” Optimization – Old Dogs
and New Tasks”, Kaplan and Otten, Mathematics Teacher, May 2012 8th grade Algebra I class
What is your immediate “gut” reaction of this problem’s difficulty? “I feel like this will be in the middle of hard and
easy. It won’t be the hardest, but it won’t be just a quick problem.”
“I think this problem will be really easy.” “I think if I kept trying, it wouldn’t be that hard.” “I think it’s easy, peazy, lemon squeezy.”
R. Lochel. ATMOPAV 2012
STRATEGY 3: EMBRACE ALTERNATE MEANS OF “TURNING IN” PROBLEMS Students work through ideas together on day
1.
R. Lochel. ATMOPAV 2012
STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST www.101qs.com, by Dan Meyer. Excellent
source of lesson starters. Teacher-contributed pictures and videos
Academic Geometry class (10th grade) Choose a picture which poses an “interesting”
geometric question. List information and formulas you will need to
solve the problem. Carry out the steps needed to solve the problem. Summarize your work using presentation software
(PPT or Prezi)
R. Lochel. ATMOPAV 2012
STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST
R. Lochel. ATMOPAV 2012
STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST
R. Lochel. ATMOPAV 2012
STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST
R. Lochel. ATMOPAV 2012
STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Contest problems often provide interesting
challenges: A 3x3 square is partitioned into 9 unit squares.
Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 90 degrees clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? (2012 AMC-12 Exam)
Worked with Prob/Stat classes to discuss problem-solving strategies.
R. Lochel. ATMOPAV 2012
STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING What is your immediate, gut reaction of the
difficulty of this problem? “This may take some effort, but I might be able to
do it.” “I have no idea what I just read.” “This is hard and confusing, but I think I can do it
eventually.” “This sucks, and I don’t want to do it.” “Huh?”
R. Lochel. ATMOPAV 2012
STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Suggest some opening steps.
“Draw a picture” “Underline and interpret difficult words”
With your team, sketch examples of squares which work, and don’t work:
R. Lochel. ATMOPAV 2012
STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING After working as a team, what has been
established? “Center has to be shaded. At least 4 others have
to be shaded.” “More than 1 way to accomplish task.” Need to “re-read and figure out what has been
said.”
R. Lochel. ATMOPAV 2012
STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Has your assessment of this problem’s
difficulty changed? “No, because this usually happens to me on tests
and quizzes. I glance at it and freak but read it over a few times and understand.”
“Yes. I have become more confident in this type of problem.”
“My opinion {this sucks} has changed. While it is a hard problem, I can do it now.”
“Very slightly. When I found a way that worked it looked easy, but as I went on it got harder.”
R. Lochel. ATMOPAV 2012
IN SUMMARY….. STRATEGY 1: PROVIDE AUTHENTIC TASKS AND
ENCOURAGE DISCUSSION STRATEGY 2: DON’T SAVE PROJECTS AND
ACTIVITIES FOR THE END OF UNITS STRATEGY 3: EMBRACE ALTERNATE MEANS OF
“TURNING IN” PROBLEMS STRATEGY 4: ALLOW STUDENTS TO SELF-
SELECT PROBLEMS OF INTEREST STRATEGY 5: ENCOURAGE THOUGHTFUL
REFLECTION AND SHARING