Challenging problems in math education and "Problems of the Week"
Have you ever wondered how a tailor or a car mechanic or a hairdresser got so skillful at what he/she is doing? Some of it may be talent and natural abilities, but a lot of it is due to EXPERIENCE or we could call it PRACTICE.
And, have you ever wondered how some people get to be good problem solvers in math? I will tell you the same thing: some of it may be natural talent, but a lot of it is due to PRACTICE. Those folks have solved many problems!
And I don't mean just simple calculation problems, but true problems that require the student to think a little.
All math curricula supposedly "emphasize problem solving". They have charts about problem solving strategies. They have lessons which concentrate on a certain strategy. But, does that sort of thing really help the students a lot? If you know what strategy you're supposed to use on a problem, then the problems are easy. Sure, it helps to see examples of problem situations. But that is just "mild exercise" - it won't yet get your student to be a proficient problem solver in math.
And kids' minds NEED 'exercise' to grow, to develop. You need to feed the mind, and you need to provide good exercise. Most of the problems in a typical math book are to help the student learn concepts and procedures. Then some are 'applications' or word problems of some sort.
Often, lamentably, the word problems are in the end of a lesson and are solved using the same operation that is studied in the lesson. This just about leaves word problems with the status "last and least".
Traits of a good problem solver are:
- does not easily give up
- is able to evaluate own progress: is this leading me somewhere or should I take another path?
- besides subject knowledge, has a vast strategy toolbox at his disposal.
All these traits are developed by solving non-routine problems.
Another, very useful activity is to OBSERVE expert problem solvers in action. That would usually be a math teacher. You see, mathematicians or math teachers don't find solutions to tough problems immediately. They stray at first, then try something else. That's just natural for any true problem solving.
But how many kids expect that a mathematical problem is supposed to be solved in 1-2 steps, immediately, without any errant pathways?? That does not happen in real life problem solving (and I don't mean just math problems).
Implementing problem solving
You could implement problem solving in math in various ways.
- Devote one day each week (or one every two weeks) to solving challenging problems - or even just one might be enough. The rest of the math lessons during the week would be left for learning concepts, procedures, and solving routine problems.
- Or, you could use a "Problem of the Week" activity. Student strives with the problem for several days, then submits a solution. There is NO answer key to look to. The struggle is there. But the benefits are substantial, as I'm sure you can readily see.
From this 'problem of the week' list you can find some that are free to use and submit to. However, in the rest of this article I want to highlight a particular "problem of the week" service that has been developed to a truly professional level:
The Math Forum Problems of the Week (PoWs)
The Math Forum PoW is a subscription service with a low price ($15 per individual student per year). The problems are meant for elementary, middle, and high school students.
Some of the main features of the Math Forum PoWs are:
- The Math Forum PoWs come in four different flavors: Math Fundamentals (grades 3-5), Pre-Algebra (grades 6-8), Algebra, and (high school) Geometry. Also available are special problem libraries for Discrete Math and Trig/Calculus.
- A student can revise his solution unlimited number of times (a fantastic feature!)
- Mentoring is available.
(There are two kinds of mentoring offered: you can either pay an extra fee for it, or receive free mentoring which is subject to availability.)
One thing I really liked about The Math Forum PoW is that it really encourages students to write mathematics! You know, we teachers always wish that students learn to write and communicate their solutions in a clear manner. Few are the students who couldn't improve their mathematical writing skills.
The Math Forum mentors use a specific scoring rubric to score a submitted solution. The rubric has two main parts: the actual problem solving, and how it is communicated (explained). Those are further divided to three parts. This sort of detailed feedback helps students more than a blank 'right' or 'wrong'. You as the parent/teacher can use this same scoring rubric if you choose not to pay extra for the mentoring, or if free mentoring is not available.
In other words, both good problem solving AND strong mathematical communication are important in the submissions to the Math Forum Problems of the Week. The student's solution should include enough information to help another student understand the steps that he took and the decisions that he made in solving the problem.
After submitting a solution, the student can then revise his solution - maybe because you as the teacher, or a mentor prompted him to think about the problem again, or maybe because the student on his own thought of a better way to write the solution.
As Suzanne Alejandre, Educational Resource & Service Developer from The Math Forum, says,
"Our research has shown that a student's mathematical thinking and their communication skills really improve if they are encouraged to revise. ... It's really a mindset -- it's that idea that you treat problem solving as you would the writing process. You have a draft, you reflect, you talk with peers or someone (like the teacher) who has read your first draft, you write a second draft, you reflect, etc."
Click on the links below to see some example problems open in a pop-up window.
As you can see, they are not your run-of-the mill problems, but non-routine and usually connected to real-life, or in some way utilize an interesting setting.
Problem of the Week from a homeschooler's perspective
Let's say you pay the $15 "Individual Membership" to get access to current problems of the week for the entire school year.
There are four problems, posted two and two on alternate weeks. The problems are open for two weeks.
As your child reads the problem, you might provide help if the language is difficult. But don't solve the problem for him! Maybe you have a tendency to help your child even with the smallest difficulties. Resist the urge. Let him think - even for a few days! (Of course, you be the judge and provide help if the problem seems unsurmountable.)
Encourage your child to first write a solution DRAFT on paper. This is not yet final - the solution writing is a process of revisions. Then you provide some feedback on this first draft.
However, DON'T just spout out, "This you did right, this you did wrong." Instead, try to get the student to think about his own solution, discover parts that are wrong on his own, or explain his thinking better. The best help and feedback is often in the form of questions. For example, "I can see that you used the '2 miles' in your proportion. Could you tell me more about the equivalent fractions and why you used the numbers that you did?"
After revisions, maybe in a few days, the student can then submit the solution at the PoW website and read the Answer Check (learn the right answer). That can prompt the student to rethink his solution again, revise and resubmit it.
After two weeks, commentary and highlighted solutions are posted on the PoW website - again a time for reflecting back and learning. And then onto the next problem!
If you buy a Class Membership, you also get access to a vast Library of past problems that is organized by mathematical subject areas. These archived problems have Commentary/Sample Solutions, for the same purpose of letting students reflect on how their solution compares to others.
The Math Forum Problems of the Week is a subscription service - not totally free - but I feel you get a great service and quality for the price. It is an excellent addition to just about any existing math curriculum and I highly recommend it for homeschoolers.
The pricing is as follows:
- Individual Membership (student): $15 per year. Unrestricted access to the four Current Problems.
- Individual Membership (teacher/parent): $25 per year. Same as above but also gives access to teacher support pages. These support pages explain what mathematical topics each problem touches, how it aligns with NCTM standards, and has links to other related resources.
- Also available is a Class Membership (for higher price). That one also gives access to two services in the Active Problem Library AND an account to monitor student submissions and the possibility of mentoring their own students using the interactive system. If interested in this option, call Lori Wall at The Math Forum at: 800-756-7823 menu option 1.
You can also get a free trial account for 21 days.