In my last blog, I discussed ways to make math more understandable. One of my suggestions was to make a habit of solving problems in more than one way. At first glance, that might seem like a silly waste of time. But I believe this will actually help you become a better problem solver. You see, to be a good problem solver, you need to make good decisions. And to make good decisions, you need to have options.
This time, I’m shifting my focus from understanding to meaning. Yes, there’s a difference, even if they are very much related. Understanding comes from having options to solve; meaning comes from making sense of solutions. As such, I will introduce you to two general strategies that you can use to explore options and make sense of solutions anytime you engage in solving math problems!
Solving Everyday Problems
Here’s a fun new word to add to your vocabulary – heuristic (prounounced “hear is-stick”). The Oxford dictionary defines heuristic as “proceeding to a solution by trial and error or by rules that are only loosely defined.” Think: Is that a good or bad way to solve a problem? It really depends. Trial and error can be seen as clumsy and aimless (bad), or as a willingness to take risks and learn from mistakes (good). Loosely defined rules could be seen as imprecise (bad), or open for interpretation (good).
Psychologists say a heuristic is like a cognitive rule of thumb (“Heuristics,” n.d.). For example, many people generally believe if something is scarce then it’s valuable. And, generally speaking, things that are “one-of-a-kind” or “original” do tend to cost more! But not always. When stores ran out of toilet paper during the pandemic it didn’t make toilet paper more valuable – at least not in the way we might treat a rare baseball card. So, “scarce is valuable” as a rule is loosely defined and only sometimes true. That is, a heuristic! They serve us in the moment, using preexisting knowledge and experience, when other information isn’t readily available.
A Classic Math Heuristic
In 1945, mathematician George Polya created a heuristic for his students to give them a generic step-by-step process to solve any math problem (Gautam, 2020). His 4-step problem-solving method is as follows:
- Understand the problem
- Devise a plan
- Carry out the plan
- Look back and reflect
Believe it or not, this same 4-step method is still used today, almost 80 years later! As a math educator, I’ve shared it with my students countless times. Whether the prompt is “What’s 34+7” or “When will an arrow traveling 20 mph hit a target 15 feet away?”, we must first understand what’s being asked, then devise a plan or strategy, then carry out the plan and, lastly, decide if the answer makes sense. For example, I understand “34+7” is asking for a sum, so I will use the addition algorithm. This requires me to regroup (4+7) as (10+1) to get 41 as my answer. That makes sense because 34+6=40, so one more would be 41. Sounds great, right?
Unforeseen Loss of Meaning
Sadly I’ve noticed, in my own classroom and in the many classrooms I observe, that most students don’t follow the 4 steps, no matter how often we model them! Instead they devise and carry out a plan, usually the one taught that day, with little understanding and without checking to see if their answer makes sense. I hope not you, dear reader! It seems that, despite our best efforts to teach this heuristic, students still rush through their math. By consequence, their strategies and results are meaningless.
Part of the issue might be that this heuristic isn’t always useful. I’ve taught step one: understand the problem by drawing diagrams, underlining words, or writing down related formulas to make my thinking visible. But I suspect this looks and feels fake to my students. My slow, meticulous modeling isn’t really how I’d solve it if I wasn’t teaching. Frankly, I don’t need a 4-step process to figure out 34+7. That type of math has been ingrained in my head for years!
Now, if the question is more complicated and non-routine, then I might genuinely take the time to write down some initial ideas to help build my understanding. Earlier I asked about an arrow hitting a target. There’s a lot to unpack in such a question before I know what my solution path will be. It feels natural to take some time upfront to get a handle on how to begin. My experience suggests then that Polya’s heuristic is great in unfamiliar territory, but not so great for routine problems that have standard solution paths.
A New Heuristic
So, here’s the issue: Using Polya’s problem-solving method for all math problems makes it look and feel fake. This causes students to not take it seriously. And that leads to fast, meaningless calculations. So how do we fix this? First, let’s agree to only use Polya’s 4-step method for solving non-routine problems that require creative thinking—this will look and feel natural and appropriate. Second, we need a new heuristic that is useful for even the simplest of problems! I propose the following 3-step method:
- Estimate
- Calculate
- Sense-make
It’s simple. It kind of rhymes. And it encourages meaningful engagement with each and every problem. Let’s call it “Krill’s 3-step meaning-making method.” Has a nice ring to it, don’t you agree? Essentially, it’s a heuristic for number sense. Number sense is “...the ability to produce reasonable estimates, to detect arithmetical errors, to choose the most efficient calculating procedure, and to recognize number patterns” (Howden, 1989). So, by using my meaning-making method we are habitually building stronger number sense.
Let’s try it with our earlier problem, “What’s 34+7?” First, estimate! This is pretty open-ended. You might say, “Well, less than 44 because we’re adding a number less than 10.” Or, “More than 34 because we’re adding.” Gosh, seems like an obvious thing to say. So what! We’re trying to break the habit of fast, meaningless math so even obvious statements move us in the right direction. Second, calculate! Use your fingers, toes, the standard algorithm, decomposing, whatever! Just do something! Third, sense-make. Just refer back to your estimate! You said more than 34, is it? You said less than 44, is it?
Notice this moves you away from the constant need to ask the teacher or check the answer key because your answer is being judged by your own estimate. It’s sense-making from start to finish! You should also know that a common error when using the addition algorithm is to add (4+7=11), write 1 down, and forget to carry the other 1 to the tens column. Oops! This would give you an answer of 31 instead of 41. Ever make this mistake? Well, guess what. Even the obvious “more than 34” estimate would save you! That’s the power of a useful heuristic.
Meaningful Math For All
Remember, I’m not suggesting Krill’s 3-step meaning-making method works for everything. Instead, I’m suggesting you use both heuristics in your math learning – Polya’s for non-routine problems and mine for routine ones. So how do you decide? If you can’t make an estimate, for whatever reason, then go with Polya’s 4-step method. What is simple or complicated, standard or surprising, really depends on experience and intuition. So don’t dwell on it. Polya’s method was a gift to mathematicians-in-training, like you, to think like an expert when the going gets tough. Use it when you need it! Otherwise, use mine. Because there’s really never a time when math should be done without meaning.
References:
Howden, H. (1989). Teaching number sense. The Arithmetic Teacher, 36(6), 6–11. https://doi.org/10.5951/AT.36.6.0006
Psychology Today. (n.d.). Heuristics. Retrieved November 22, 2024, from https://www.psychologytoday.com/us/basics/heuristics
Gautam, S. (2020, March 5). Four steps of Polya’s problem-solving techniques. Medium. https://medium.com/enjoy-algorithm/four-steps-of-polyas-problem-solving-techniques-80eb39d51c94
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