I am convinced that improving problem is possible within the context of
virtually any math lesson.  I would like to hear responses to the comments
below from other math teachers.    Sue Abrams, VISMT

At the heart of improving problem solving is providing ALL students with
opportunities to engage in problem solving on a daily basis.  While we all
recognize the importance of having our children solve rich open-ended
problems such as those we use as portfolio tasks, we may not recognize that
can help develop problem solving skills in our students EVERY DAY  if we
create an atmosphere in which ALL students regularly

    • make their own observations, however insignificant (and hear the
    observations of classmates)

    • probe their own reasoning (and that of others)

    • make and test their own conjectures (and test the conjectures of
others)

    • synthesize the big math ideas that are at the heart of each lesson


Specifically, we classroom teachers can make each day rich in problem
solving
if we do the following:

1.  Have students write -- or in some way note -- observations before
sharing
with a partner and/or the whole class.

2.  Pause...twice (once after asking the question; then once after an answer

is proposed)

3.  Avoid praising correct answers or even saying "right"  (This does NOT
mean letting students think that any answer is OK, however.)

4.  Model problem solving ourselves.  (First I noticed that...; then I
tried...., but I knew it wouldn't work when....; so I decided....)

5.  Have regular experiences where each child synthesizes his/her learning.

(reflective writing, think/pair/share, end of class "mathematicians circle")

6.  "Un"scaffold the lesson to create richer problem solving possibilities
(more important in some published programs than others)


7.  ASK "GENUINE" QUESTIONS and MAKE COMMENTS THAT INVITE DEEPER THINKING
(Keep in mind that DISEQUILIBRIUM is needed for meaningful new learning to
take place.)

• Why? / Why not?

• Help me understand ...

• Tell me more ...

• Hmm ...

• Convince me ...

• What do you notice / observe?

• I see that you look (puzzled).  What are you wondering about / thinking?

• What if...? / What if ... not ... ?

• We have (several) solutions here.  I am confused ...

• When you said / did ..., tell me what you were thinking.

• How is this like / different from...?

• Will that always work?  Why / why not?

• What questions do you have for (John)?

• What are your questions?  (Then answer with a question.)

• Describe what YOU think (Beth) is saying, and ask her if you are correct.

• (Jenny), perhaps you can help me understand what (Joey) is saying.

• How many ways can you ...?