Thursday, February 5, 2015

Top 10 Worst Problem-Solving Habits

As promised, I am back with a new post to share the worst problem-solving habits and ways to get "unstuck" when trying to solve story problems.  Below is a list of the Top 10 worst habits:

          1.  Trying to do a problem "all in your head" and not writing anything down.

          2.  Arbitrarily choosing a strategy.

          3.  Staying with a strategy when it is not working.

          4.  Giving up on a strategy too early.

          5.  Getting fixated on a single strategy and trying to use it for everything.

          6.  Not asking yourself:  "Does this make sense?"

          7.  Being afraid to ask for help.

          8.  Not checking your answer.

          9.  Not noticing patterns.

         10.  Going through the motions instead of thinking.

I am sure if I asked my former math students they could probably recite many of these from memory...they definitely heard them often.   Hopefully, they all also learned the best ways to get "unstuck" (or avoid getting "stuck" in the first place!)

So if you are "stuck" one of the following may have occurred:

          Perhaps you tried to rush through the problem without thinking
          You didn't read the problem carefully.

           Maybe you don't know what the problem is asking for
          You don't have enough information

           Could you be looking for an answer that the problem isn't asking for?

          Sometimes the strategy you're using doesn't work for this particular problem
          You aren't applying your strategy correctly
          You failed to combine your strategy with another strategy.

          Finally, the problem could have more than one answer
          Possibly, the problem can not be solved !!!

If any of the above happens to you while trying to solve a problem, here is what you do--

          Re-read the problem.  Then modify, change or combine your strategy with another.
          Look at the problem from a new perspective or try looking at other similar problems.
          Get help and wait awhile and try again.

And remember, to be a good problem solver you need to...
          Solve lots of problems.     Learn from the answers.    Use whatever strategy works. 
                        Watch other people solve problems.          Rework problems.    
          Seek help and ask questions, and don't be afraid of asking" dumb "questions.

I am happy to report that my students are doing a wonderful job learning and practicing problem solving strategies.  In fact, your son or daughter may have shared with you that they are starting to write their own logic problems!!


Monday, December 15, 2014

What Good Mathematicians Do

Mathematics skills are an important element of our everyday lives and there are certain behaviors that young mathematicians can practice to help them solve problems both in the classroom and in their daily lives.  Of all the subjects students learn in school, math is the most predictable (which is one of the reasons I love it!).  Often you can find a solution to a problem by identifying a pattern and continuing it.  Young children learn to count by twos and later learn that multiplication facts follow a pattern.  Pascal, whose work helped design computers and Fibonacci, whose work helped explain events in nature, were early mathematicians known for patterning.

In math, "a picture can be worth a thousand words."  Drawing a picture or diagram based on information contained in a word problem or putting numerical information into chart form allows a student to organize data visually.  Students should practice with venn diagrams as well as bar, line and pie graphs.

All good mathematicians estimate.  Estimation is not wild guessing but an important math skill that requires thoughtful consideration.  Everyday we  estimate when we choose a pitcher large enough to hold the lemonade we are preparing or find a box big enough to hold all the items we want to pack.  More importantly, estimating is a method of quickly assessing whether an answer to a math problem makes sense.

Students learn valuable math skills when they ask questions.  Of course, the first question they usually ask is "Is this right?" but as they mature they wonder "Are there other ways to solve the problem?" or "Is this always true?"  When students work in groups, they should ask "How did you find that answer?" or "Why did you solve the problem that way?"

Good mathematicians create a plan to solve a problem.  Allowing for think time before you begin to solve a problem is an important skill.  "Plan your work and work your plan" is a good slogan for every math classroom.  We want to focus both on the answer and the path we take to find it.  We should all understand that it's okay to change our plan and take a different direction when a strategy isn't working.

To grow as a mathematician, a student must be able to explain their work.  The teacher's method of solving a problem is not the only accepted way.  A student could have arrived at the correct answer using a different method or the problem solver may discover an error while talking about how they solved a problem.  It is also imperative that math students check their work.  Convincing children to go over their work to look for careless errors is quite a task...they simply want to finish.  But good mathematicians take time to do a good job.  Math students often seem in a hurry to finish a math assignment.  In addition to not taking the time to check answers, they might give up easily or guess when the answer doesn't come right away.  If a problem has them perplexed, they should move on and then revisit the problem later.

Good mathematicians seek new ways to find a solution.  The more ways students learn to find a correct answer, the more they stretch and grow mathematically and intellectually.  As teachers we should accept and encourage a variety of problem-solving methods and our students will enjoy demonstrating new ways to find answers!  In addition, requiring students to solve a problem in more than one way is a roundabout way of requiring them to check their work.  If two problem-solving methods result in two different answers, students know they need to review their methods and calculations.

And finally, good mathematicians know they may use technology as an aid.  Mental math or pencil and paper can provide quick answers to some problems.  In some situations, however, especially when figures are large or accuracy and speed are vital, the use of a calculator or computer is the best way to find a solution.  And of course, computers and tablets can be used to find interesting math problems, math games and online math manipulatives.

In my next blog, I will share the "Top 10 Worst Problem-Solving Habits" and ways to get "unstuck"
when trying to solve story problems.

Wednesday, November 19, 2014

Calculators and Problem-Solving

It is time to explore the last two components of my class...calculators and problem solving.  These make a great team when calculators eliminate the drudgery of lengthy computation so students can concentrate on the problem solving.  After all, the manipulation of numbers is not the prime objective in math...we need to have the ability to apply the appropriate function...we must know when to multiply and subtract as well as how to multiply and subtract.  They also must know what information is required and whether an answer is reasonable.  The calculator, when used as a tool and not a substitute for computation, will free students to spend more time thinking and less time writing.

So far, students in my class are getting to know their calculators and have learned the various keys and features.  The children are particularly fond of "Calculator Codes"--this is a self-checking system because the answer shown on the display spells a word.  The numbers, when viewed upside down on the display, become letters.  I'm sure if you ask your child they can show you an example.

The final component, which we are just starting to explore, is the most important.  Math skills are an important element of everyday life and students should understand they use math to complete a variety of tasks each day.  There are many strategies we can use to solve problems listed below.

                                                      1.  Look for a Pattern
                                                      2.  Make a Picture
                                                      3.  Use Logical Thinking
                                                      4.  Make an Organized List
                                                      5.  Guess and Check
                                                      6.  Make a Table
                                                      7.  Solve a Simpler Problem
                                                      8.  Work Backwards
                                                      9.  Exclude Extra Information
                                                     10.  Find "assumed" Steps
                                                     11.  Use Functions
                                                     12.  Use Venn Diagrams
                                                     13.  Use Logic Squares
                                                     14.  Solve Multi-step Problems
                                                     15.  Use Data from a Chart
                                                     16.  Solve Multi-answer Problems

Students will also practice recognizing the "clue words" which signal what operation they should use when solving a word problem.

Problem solving is a skill that can be developed and the more problems you solve, the better you get and the more fun problem solving becomes.  Almost all problems can be solved using the strategies listed above and depending on the situation, some strategies work better than others.  But there is also no single "correct" strategy for solving a problem and any strategy that works is the correct strategy!

In the coming posts to this blog, I would like to do a more in-depth examination of problem solving.  We'll look at what good mathematicians do as well as the worst problem solving habits.

Tuesday, November 4, 2014

Analogies and Logic

So, as promised, I am back to describe two other components of the class I am teaching this year to our first through fifth graders.  If you missed my first blog, you might want to go back and read about Brainstorming and Brainteasers.  This week, I want to focus on Analogies and Logic problems.

An analogy is a comparison between two things.  It points out the similarities between two things that might be different in all other respects.  Analogies can be both visual and verbal.  In order to solve analogies, students need to analyze the elements of the puzzle, define the relationship of two things, and mentally compute a similar relationship between two new things.  The children really learn flexible and critical thinking as they look for creative combinations.  Sometimes they find combinations that are not as obvious but still merit consideration, especially when the student can explain their line of reasoning.  What is most valuable about this is when children see that developing their reasoning abilities is more valuable than just seeking out one correct answer! Analogies also build vocabulary skills.  Some of the types of analogies include synonyms, antonyms, homophones and rhyming words, characteristics, composition, general/specific, groups and parts to name a few.
I actually anticipate that your children will be able to write their own analogies by year-end!

Here's a few analogy examples:  p is like q as b is like  p, d or g

Word is to sentence as letter is to  symbol, word or write

One of my favorite things in life is solving logic problems.  This year we will work with logic problems on a grid or matrix, on a venn diagram, and complete number and poster logic.  So far this year, we have used grids to organize information and reach conclusions.  Working in groups, students read a paragraph which sets the background for the puzzle.  A list of "clues" follows, and by sorting, eliminating and associating, the grid is completed.  Many times, with just one clue, several eliminations or positive connections can be made.  I am amazed at how quickly our students have moved to more difficult levels of deductive thinking!  Below  is an example of a problem my second graders can easily complete!

In my last blog I promised to provide the names of some free apps that I will be using.
 This is to That is a pop culture analogy game.  You can also try Kids i Help - Analogy 1.0, 2.0 and Word Analogy .  I will introduce these to the students this week so they will be familiar with the way they work.

Next time, I will describe the last two components in my class--calculators and problem solving strategies.  I hope your children are enjoying thinking and learning in my class as much as I enjoy teaching it!

Tuesday, October 21, 2014

My First Blog!

Now that we have wrapped up the first quarter of the school year, I wanted to brag about our students who are by far the best and brightest I have ever had the pleasure to teach!  I would like to spend the next few weeks writing about the various components of my classes with Grades 1-5.  I named my class Problem Solving and Logic but it is really so much more.  It is divided into six parts…Brainstorming, Brainteasers, Analogies, Logic, Calculators and Problem Solving Strategies.  Let’s take a closer look at the first two parts.

Brainstorming emphasizes the skill of fluency by generating a large number of ideas or alternative solutions to a problem.  Students really never give a “wrong” answer when they are asked to brainstorm and I have noticed an enormous amount of originality as they try to give a unique or unusual answer…an answer which no one else in the class might have considered.  Brainstorming activities have also led to the use of venn diagrams—ask you child to show you!

Brainteasers have many forms ranging from hidden meanings, paradoxes, idioms, visual tricks and common attributes.  What began as me bringing one or two to each class for the students to solve has mushroomed into your creative sons and daughters making their own and hoping to stump their classmates (and teachers!!)

Another facet to my class this year is that I am attempting to, as much as possible, make it paperless.  (You are probably thinking that means we are using computers or mobile devices which is not the case, although as we move through the 2nd quarter I will be telling you about some great apps.)  Instead we have utilized individual dry erase boards and pockets.  There has been an emphasis on working in small groups to promote flexibility.  Seeing things from different points of view or solving a problem in different ways while listening to one another helps students learn.

My goal is to help your children become motivated and successful problem solvers with activities that are fun, engaging, challenging and stimulating.  Look for more information on Logic and Analogies next week.