Differences between exercises and problems: Task 2-3

[Maria Droujkova]

Hello,

I would like to stage a discussion about the topic that interests me very much: exercises vs. problems.

You will find many conflicting definitions of what problem solving means. Here is one I use, to contrast problems and exercises:
- Solving a problem, you do not know ahead of time what methods and areas of math to use. Solving an exercise, you know exactly what methods and areas of math to use.

This definition can use much improvement, though. Comments are welcome about any of the following:

Do you have a good definition of a math problem?
Can a teacher know, creating a set of tasks for a group, if they are problems or exercises?
If you could tailor your math content really carefully, what is problem:exercise ratio you would use in your student tasks? Is it the same for every student?
In one controversial book claims that problem solving is most beneficial for populations that struggle with traditional math, namely low socio-economic status students, minorities, and girls. What do you think of problems vs. exercises for weak and strong students? Reference: J. Boaler, Experiencing School Mathematics: Traditional and Reform Approaches To Teaching and Their Impact on Student Learning, Revised and Expanded Edition, Revised. (Routledge, 2002).

Cheers,
Maria Droujkova

Make math your own, to make your own math.

 

[Doug Snyder]

Maria -

Great questions for discussion!

When I think of exercises, I think of "Open to Page 75 in your math book and do problems 1-12".  That statement would be preceded by some notes and some guided examples ... "Recipe Mathematics".

When I think of problems, I think of problems 95-100 in the same section ... the problems that we don't usually get to in class.  Or, I think of the "Open Ended" questions that are on the PSSA exams, which is Pennsylvania's standardized testing.  For many of the students in my school, our students do not even attempt to answer the Open Ended questions.

My definition of a math problem:  A real-world problem requiring application of mathematics for some part of its solution.

Based on the first part of my post, I believe it is not difficult to to know if tasks are problems or exercises.  Call me "old school", but I believe that the repetition of math exercises is important for our students.  Doing multiple exercises continually reinforces the concepts being learned.  However, I believe that doing problems is where many of our students "shut down".  I believe that Dan Myers made some excellent points about our students not wanting to take the effort needed - in terms of time and brain power - to solve "real problems". 

In many of my classes, I have made the "problems" either a once a week exercise or extra credit for my students.  The "once a week" scenario are what I call "PSSA Days", which are mandated for every math class in our school.  One day per week is designated for focusing on PSSA preparation.  I have chosen to make these days more and more problem solving days.  I have differentiated between my students' abilities by allowing students to work in small groups to complete this work or by giving some classes, or some groups, a smaller number of problems to solve.  I would say, however, that these days are difficult for my students because they truly struggle with problem solving, in general.

[Al Williams]

Definitions
Solving a problem: You have access to the pantry of ingredients. Make a cake.
Exercising: Make a cake by mixing two cups of flour, 2 eggs, and one
stick of butter. Mix thoroughly. Put in 6 by 12 pan and bake at 400
degrees for 1 hour. Let cool. Cover with frosting.

Can a teacher know, creating a set of tasks for a group, if they are
problems or exercises?

Better question would be how the teacher presents the problem to the
group and whether he or she will grade the problem correct by
following a specific set of steps. If the teacher creates the tasks
for a group, then it is an exercise. If the teacher marks it correct
only if the specific steps have been followed in a linear fashion,
then it is an exercise.

If you could tailor your math content really carefully, what is
problem:exercise ratio you would use in your student tasks? Is it the
same for every student?

I think it depends on where the students are. If you follow the "I
do, we do, you do" mantra of lesson planning, I think exercises are
okay for the I do and we do component. I think, then, that problem
solving would be more appropriate for the we do and you do components.
And, to really help with problem solving, I think the I do component
should show several paths of steps that could solve the problem. For
high structured classes: exercises 70% problems 30%. For low
structured classes: exercises 30% problems 70%.

In one controversial book claims that problem solving is most
beneficial for populations that struggle with traditional math, namely
low socio-economic status students, minorities, and girls. What do you

think of problems vs.exercises for weak and strong students?
I think that minority status and gender status do not matter. If a
child comes to school prepared to learn and with a desire to learn,
then it doesn't matter. They will learn with both methods. Of
course, I would propose that they will learn to think better with
problem solving. I'm not yet sure how much socio-economic status
correlates with a student's learning. I'll be curious to hear what
others think.

[Ethan Lewis]

Maria, I like your definition since it is so succinct

"- Solving a problem, you do not know ahead of time what methods and areas of math to use. Solving an exercise, you know exactly what methods and areas of math to use."

At the risk of complicating things, I will search for another angle:

How about a comparison to how exercises are used in some some physical activities, say music or athletics.

In these activities exercises (or drills) are used to "drum in" simple skills that are basic building blocks for more complex activity further down the road.  The idea is that these activities will become automatic and, ideally, be done accurately without conscious awareness.  Only with these skills in place can a practitioner in athletics or music really excel, as her/his mind is free to grapple with (and enjoy) the complex activity that is the real source of interest in his craft.  For example most people who take up music are interested in playing songs or more complicated works.  However, in order to master these works, it is necessary to spend plenty of time working on exercises such as scales. 

I see mathematics in the same vein.  The real enjoyment and goal of math is in problem solving.  Exercises and drills are activities to support problem solving.

Do you have a good definition of a math problem?

  Here's a try: It requires complex activity with more than one step.  Solving a problem should require more than an immediate application of some recently learned formula or technique.

Can a teacher know, creating a set of tasks for a group, if they are problems or exercises?

  I think you can only make an educated guess.  The approach to solving exercises should be obvious.  As exercises increase in difficulty they become more like problems, since how to apply the recently learned skill is no longer obvious.

If you could tailor your math content really carefully, what is problem:exercise ratio you would use in your student tasks? Is it the same for every student?

  Definitely not the same for every student, since what are exercises for one student may be problems for another.  To do this (ratio) really well, it needs to be individually tailored.  The goal is to develop independent problem solving skills, but I don't know what the correct ratio is.  I hope I'll have more to say about this after I've had experience teaching.

What do you think of problems vs. exercises for weak and strong students? 

  I like the idea of a problem solving approach for weaker students.  The issue is to adjust the difficulty of the problems to a suitable level for the students.

Ethan

[GASS2]

Problem solving requires critical thinking, while exercises require
specific skills. A teacher knows if the task set for a group is an
exercise or a problem. With most exercises there is a procedure to
follow in arriving at the correct answer. Problem solving has several
ways of fixing the problem. The student is allowed to be creative and
the mind is stimulated in problem solving. Exercises is repetitive and
if not careful can be boring.
The problem vs exercise ratio requires the teacher to know his/her
students and be able to motivate for higher learning. I grew up
learning math the traditional way. I know that I would have benefited
even more from a bit of diversity. Technology has advanced so much,
that we can no longer lean towards more exercise. We must advance, by
allow the mind to expand beyond the norm. We must realize that
analytical thinking goes beyond the classroom. Exercises just helps us
to pass from one class to another.

Grace

[Amy]

I agree with many of you. Exercises are work we do with the students
after teaching them a specific formula, strategy and we want them to
practice it. I think this is what math is to many of us. Opening a
workbook, memorizing a formula, applying the formula to solve the
problem. Problem solving to me is more open-ended and not so "cookie
cutter" where you just plug in a formula. Like Doug said, it is more
of a real life issue that you need to use mathematics to solve.

I think a lot of teachers shy away from doing more "problem solving"
and do more "exercises" because it is quick, to the point, and it's
usually laid out for us in the math books. Having kids do open-ended
type problems is very valuable but it takes a lot of time, time we
don't always have. I feel like if we spent more time on problem
solving it would be at the expense of other topics that we can't just
skip over. It hard to squeeze everything in!

I think that all kids should be exposed to both exercises and problem
solving. I think I would do more of the exercises in the beginning
because it is important to learn the formulas and practice applying
them to solve problems. I think it becomes less effective when you do
exercises all of the time after the kids have become comfortable with
it. I think the kids need to be challenged when they are ready for it.
This is what we call Differentiated Instruction.

I think that many kids prefer "exercises" because usually there is one
definite answer to the problem. It is either right or wrong. When it
comes to problem solving there might be many different answers,
therefore not a concise right or wrong. There are many variables and
possibilities with problem solving and I think that scares kids (even
adults) to have to start from scratch and come up with so much on your
own with out a crutch/safety net to guide you.

[Gillin, Kathryn]

Everyone has said a lot already about problems vs. exercises. It seems like most agree that exercises lean toward the rote practice, while problems involve a little more critical thinking. So I am going to go to the topics that relate to me most here: ratio of problems to exercises, and problem solving with weaker students.

Here's the problem for me when it comes to problem solving: I have 7 students on 5 different math levels (with 5 different textbooks) and 45 minutes of math instruction a day to teach all 7 of them at the same time. If that's not a math problem, I don't know what is! I am expected to teach these students what is pretty difficult math curriculum for their ability level. Giving problems as opposed to exercises can be stressful for me and for them. My students of course make the most obvious progress from exercises. While I know it is VERY important for them to problem solve, the fact of the matter is when dealing with difficult problems they don't have the skills to do it. These students are in the bottom 5% of their peer group, most are in the bottom 1%.

How do I help them develop the skills (and confidence) to problem solve? I practice problem solving with problems that are 100% within their ability range (which, trust me, can be hard to find). We'll take an easier problem that they have mastered the skills for, and talk out how to solve it. My students are TERRIFIED of problem solving. They are middle school students on a third grade or lower reading level. They have a hard enough time figuring out what the question is asking, forget finding the answer.

While problem solving may not take up 50% of my math class, it does make its way into my room. I spend a lot of time working on problem solving with my students that is more applicable to life and social skills (i.e. I lost something in the cafeteria, how do I go about finding it?) A big part of my job is stepping back and asking myself, "Is this applicable to my student's life? Is this something they will truly use and benefit from?" Those questions direct which way my instruction goes.

If I was in a different classroom, my opinion on this would be drastically different I am sure. I am sorry I strayed from the topic a bit, but this is what problem solving looks like in my room.

Thanks!
Kate

[Gail Rice]

In response to Problem-Solving vs. Exercises:
Whenever I look a math textbook the section labeled exercise is
usually a set of 10 to 20 math problems that follow an explanation, a
formula or a definition. Under the section labeled Exercise is another
section labeled Problem-Solving and those problems usually have a
story or scenario which will require students to think about how they
will apply the skills they practiced in the previous exercises to come
up with an answer.
I think a teachers know when they are creating problems or exercises
because of the structure of the task. However they might use the
incorrect label (Exercise in stead of Problem-Solving) when referring
to them.

In order for of students to successfully solve a problem they need to
have practiced the exercises. For example, exercises in finding area
of figures will help students to recall how to find the area of a
figure when task requires that they find the area. Both problem-
solving and exercises are equally important and should have the same
ratio. The ratio of problem-solving:exercise per student should be
determined based on a student's needs and ability.
We live in a world of standardized assessment and all students need to
know how to problem solve because a portion of their assessment is
being able to solve problems and that section is heavily weighted.

On Jul 20, 4:42 pm, Maria Droujkova <droujk...@gmail.com> wrote: