Math Problems with Lots of Paths from A to B: Open Middle

Directions:  Fill in the boxes using the whole numbers 1 through 6 to make the largest (or smallest) possible number. Use each digit at most once.

       Who ever said that there is only one way to get the answer in a math problem?  

       OK...(unfortunately) lots of people say this.  But, guess what, the best math questions don't have just one way to be solved.  Because the best math problems don't have the simplistic goal of merely following the steps you are told to follow and getting an answer.  The best math problems have much broader goals: Thinking, Reasoning, Problem Solving, Perseverance

       I've recently been introduced to a great source of these math problems called Open Middle.  Open Middle math problems start the same and end the same (with one particular correct answer), but they have an "open middle"; there are many ways to approach and to ultimately solve the problem.  The above problem is a sample of an Open Middle problem.  

       Open Middle has math problems for grades Kindergarten to high school mathematics.  They are great for getting students to use recently learned math in novel ways.  And when students have a chance to talk about the way they went about solving the problem, they gain an ever stronger grasp of the mathematics by hearing from their peers--a strategy known as Number Talks.  

       Students learn that it is OK to think about solutions differently.  One student may have a very complicated solution that makes sense to her.  Another may have a clever solution that no one ever thought of--not even the teacher!  Learning from each other is a great way to experience math and to understand that math is vibrant and alive with many possibilities.

       Students may think of Open Middle problems as puzzles, but listen to the math vocabulary that they use when the work on their solutions.  Students who are bored with the typical math class are excited to spend lots of time working on Open Middle problems because they are relatively easy to start.  Every attempt at an Open Middle problem helps students to learn what to do and (perhaps) what not to do.  They don't view wrong answers as failures but instead as a closer step toward the final answer.

Directions: Using the whole numbers 1 through 9 no more than once, create 3 equivalent fractions.

Improving Education is Harder than Going to the Moon

       I often hear people say, "We have been able to send men to the moon.  Why can't we improve education?"  The implication, of course, is that the effort to send people to the moon is certainly more difficult than educating children.  So if we could do that, why can't we accomplish this--much easier--task?

       I would contend that improving education for all students is much more difficult than--the extremely complex of task of--sending people to the moon because (even today) educators are dealing with many more "unknowns" than NASA had in 1969.

       I doubt that anyone at Mission Control watched the Apollo capsule heading toward the moon thinking, "I'd say that we have a 20% chance of missing the moon by a hundred thousand miles."  They knew how far away the moon was; they knew the amount of force needed to break out of the earth's atmosphere; they knew how much food the astronauts needed.  It was certainly a complex task.  Mistakes were surely made and some things certainly went wrong.  But the Apollo 11 team that dealt with the physics of sending people to the moon were dealing with plenty of known quantities.

       In contrast, improving education involves our--still evolving--knowledge of how students learn.  Brain research is currently taking place and reporting new findings every year.  We have to contend with understanding the best way to motivate students and the best way to teach students.  And, of course, students aren't robots with the same abilities and the same limitations.  We have students from rich and poor families with different views of the benefits of education.  We have students from different home-life situations in which some are very supportive and some lack any sort of structure.

       It would make more sense to compare the effort to improve learning with the effort to cure cancer.  Over the years and decades, both of these fields have experienced progressed, but neither of these fields have been able to claim a complete victory.  Our understanding of cancer cells has certainly improved from 100 and 50 and even ten years ago, but we don't know enough to know how to stop their growth throughout the body.  Similarly, our understanding of how students learn has improved, but we are yet to find a school model--or even an education model--that best addresses the needs of all of our students.

       Improving education is a complex problem.  We see small improvements from time to time; and we see isolated pockets of (what appear to be) great success from time to time.  But we still have more to learn before this problem can be solved.  We have to continue to build on the successes of the past (and present) to reach the day when all students will receive the full education that they deserve and all students will reach their full potential.

Thinking about Thinking

       There has been a lot of talk recently about the need for schools to teach students how to think.  Actually, the need to build thinking skills has always been a big part of schooling, but our changing world seems to have added to the value of thinking skills.

     So this leads to the obvious question, How do you teach students how to think?  How do teachers create a learning environment in their classrooms that encourage students to think and to reason and to build on past knowledge?

       It is clear that there are times in a typical student's day when they are merely doing what they are told to do.  "Copy the notes,"  "Read this passage."  "Answer the questions."  Very little thinking is needed to complete these tasks.  Quite possibly, no thinking is needed to complete some of these tasks.

       So, once again, How do you teach students to think?  Or maybe I should ask, How do teachers encourage students to think?  One way to do this is to present students with problems to be solved.  We may suggest certain tools to solve the problem, but we do not tell them how to solve the problem.  The "problem" could be lots of things.  Here is a short list of problems to consider:

1. x+27=51
2. Run a mile in at least 30 seconds less time than you did at the beginning of the school year.
3. Learn to drive a car.
4. What can you do at your school to encourage students to be more friendly with each other?"

       The first "problem" could require thinking or it could require very little or no thinking.  If a tells a students to "subtract 27 from both sides of the equation", the student only needs to do-what-he's-told.  No thinking required.  If, instead, the teacher asks the question, "What number added to 27 is 51?", the student needs to do some thinking.  Of course we want the students to get the right answer, but the more important skill that we (eventually) want is for the student to know how to get the right answer.  We want to know what the student did to get the right answer--or the wrong answer.  We want the student to understand what is being asked and to devise a path to a solution.  (Lots of thinking!)

     The second "problem" is a physical problem.  Let's say that this student ran as fast as she could the first time she ran the mile, and now she is asked to run the same mile in at least 30 seconds less (later in the school year).  Let's also assume that this student has the desire to solve this problem.  She has to think, How can this be done?  What can I possibly do to run faster than my fastest running speed?

       The third problem requires a lot of procedural knowledge and a lot of rules to know.  Before students learn to drive, they often have a great deal of experience riding in a vehicle and (perhaps) observing someone driving.  This is knowledge that can help them when they are learning to drive.  They also (probably) have to learn things that they could not learn via mere observation.  Driving requires a lot of decision making which requires thinking.  Eventually experience with driving lessens the need for most of these thinking opportunities.  But in the beginning, driving requires a lot of thinking.  Considering that so many young people are able to acquire a driver's license, it must be true that even difficult problems that require a lot of thinking can be accomplished when we are sufficiently motivated to accomplish them.

     Finally the last "problem" is a social problem that is not (usually) solved through the learning of simple and procedural steps.  Indeed, this is the sort of problem that tests our abilities to think.  There is certainly more than one answer to this problem, yet these multiple answers may be very difficult to discover.

       Who knows how many difficult problems our students will face during high school, after high school, and well into their adult years?  The ability to think and to reason is a necessary asset in our lives.  Thinking can be taught and thinking can be practiced in our schools.  But thinking must be encouraged.  We cannot allow our students to go through the entire school day--everyday--and never ask them a difficult question that requires thinking.  We cannot allow them to spend all day in school merely doing-what-they-are-told.

       Students need opportunities to struggle with a problem in a classroom learning environment that provides encouragement.  The joy of thinking through a difficult problem and finding a possible solution is an experience that we want our students to have on a regular basis.  Thinking requires practice.