I am a computer educator, a math educator, and a writer. As a computer educator, I founded the non-profit International Society for Technology in Education more than 40 years ago, and headed this organization for many years (ISTE, 2020, link). As a math educator, I have served on the Board of the non-profit Math Learning Center since it was established more than 40 years ago (Math Learning Center, 2020, link). As a writer, I have authored, co-authored, and/or edited more than 60 books and about a thousand articles. Most of my writings are available free through Information Age Education, a non-profit company that I established about a dozen years ago (IAE, 2020, link). Throughout my professional career, I have been particularly interested in studying and teaching problem solving.
During these current stay-at-home days, I frequently find myself browsing the Quora website (Quora, n.d., link). The site posts a wide variety of interesting and challenging questions and problems, and it shares some of the answers provided by readers. Sometimes I submit an answer. I was somewhat surprised recently to see that an answer I submitted awhile back had received 26 positive votes. (My goodness, such fame! All of my years of being a math, computer science, and education professor are finally paying off.)
Here is the Quora question and my answer. [Note: In the remainder of this blog entry I will use the term “question” to mean either a question or a problem. I will not attempt to differentiate between a question to be answered and a problem to be solved.]
Quora Question. You walk into a room and see a bed with 2 cats, 4 dogs, and 1 cow on it. You see 2 penguins flying over it. How many legs are on the floor? (Quora, 5/23/2020, link).
My Original Answer. The answer could be as little as zero. You, the person walking into the room, may well have feet on the floor, but I doubt if your legs are on the floor. Some beds do not have legs. For example, do you consider roller casters to be legs? Also, a mattress sitting on box springs that are sitting on the floor is a bed, and has no legs. Of course, none of the animals on the bed have legs or feet on the floor. Finally, I have never seen a flying penguin, except in cartoons. In any case, a flying penguin would not have any feet or legs on the floor. [Comment added 6/15/2020: I checked the Web to see if any type of penguin can fly. The answer was “no”.]
Hmm. I wondered where the cow came from and how it got onto the bed. I didn’t think about the possibility that the cow came in with some flies on it, and that maybe there are one or more flies now standing on the floor. Also, perhaps a cat or dog has fleas, and some fleas have jumped onto the floor. How many legs do flies and fleas have? I looked this up on the Web, and they each have six legs. I also thought briefly about people, animals, etc., perhaps having one or more missing legs, but dismissed that possibility.
Quora often asks puzzle-types of questions that contain some numbers. My math background causes me to pay special attention these questions. My first reaction to any such question is to attempt to see if it makes sense. Is it a clearly stated, unambiguous question? When “perhaps not” is my tentative answer, I immediately begin to explore why the question might not make sense. That is what I did in producing my Quora response that received 26 positive votes.
What do you do when you are faced with a question that you know contains incorrect or even blatantly false information, such as flying penguins? It seems to me that it makes no sense to try to answer a question that includes information you know to be incorrect. The question may well contain additional incorrect or missing information that you have not detected.
This is a common approach to presenting fake news (Farmer, 5/31/2018, link). The fake part is mingled in with a number of other statements that may (or may not) be correct. However, any conclusion based on the information presented may well be incorrect because of the one or more parts of the information that are incorrect. Helping our students learn to identify fake news can and should be an important educational goal at all grade levels.
More About Sense Making
If I decide to go ahead and determine an answer to a Quora question, one part of my brain continues to do sense making—but of different sorts. For example, does it make any sense for me to spend my time trying to answer the question? Perhaps I find I have no interest in the question (no intrinsic motivation to try to answer it), and have better ways to spend my time. Perhaps the question is one that is easily answered by use of the Web. Perhaps the question requires knowledge and skills that I never had or no longer have. Perhaps the question has no answer or has many possible answers.
This bolded statement raises a very important issue. Even if a question (or, problem) is clearly stated, that does not mean it is answerable (solvable). So, one very important aspect of a good education is learning that not every question can be answered and not every problem can be solved. I assume that you know this. Where in your schooling, or outside of school, did you learn this? Do you think this is a topic that should receive more emphasis in formal schooling? I definitely do!
Now, suppose I go ahead and start working on the Quora question. If I succeed during the time I am willing to devote to the task, that is great. However, how can I tell if I have succeeded? In math education, students learn a variety of ways of checking their answers. This ability to check an answer, and why it is so important to do so, is another topic worthy of discussion in every subject area taught in schools.
Suppose I work on a question, but do not succeed in answering it. When should I stop trying? I am reminded of the quotation from W. C. Fields, a famous American comedian, actor, juggler, and writer, “If at first you don't succeed, try, try again. Then quit. There's no point in being a damn fool about it.”
The Quora Question and Math Education
What does the Quora question about legs on the floor have to do with my interests in math education and in computers in education? Suppose the Quora question were to be presented in a typical 4th or 5th grade math class. Many students would likely start calculating, figuring out that 4 dogs have 4 x 4 = 16 legs, two cats have 2 x 4 = 8 legs, and so on. They would then add up the number of legs they are able to identify in order to come up with a proposed answer.
Students would take this approach because much of the math they are learning is of the “brains turned off” variety. They are not learning sense-making, number sense, and general approaches to solving math and other problems. Instead, many students are learning that whenever a problem presented in a math class contains numbers, then their task is to figure out what combination of addition, subtraction, multiplication, and division will possibly contribute to solving that problem.
How many students would consider asking whether the bed has legs, or if it could be a waterbed, or perhaps just a mattress on the floor? And, what about the flying penguins? Would some students know that penguins don’t fly, and that fact makes this a very strange problem situation? It is important that we help students learn to critically examine every question and problem they are being asked to study or solve in all of the courses they are taking.
My friend Bob Albrecht and I collected examples of math problems that don’t pass our sense-making examinations. You may enjoy reading these examples (Albrecht & Moursund, 12/24/2018, link).
Most people tend to think about schooling in terms of the specific knowledge and skills that students gain. They tend to forget that much of this content will be lost over time, unless routinely used. This fact raises an important question. What can and/or should students be learning and doing in school that will be of long-term or even lifetime value? What balance should schools seek between those long-lived types of learning tasks versus time spent on learning specific content and skills that may likely soon be forgotten?
Well, here is a math education idea to think about. We have computer programs that can rapidly and accurately carry out all of the types of calculations taught in precollege math classes. So, one way to accomplish such a calculation is to “ask” (or, should I say “tell”) a computer to do it for you. That is:
- You recognize that there is a particular type of operation (arithmetic, symbol manipulation, graphing, etc.) that you want to have done.
- You have the knowledge and skills to ask a computer to do the task for you.
- You understand enough about the problem you are working on to be able to determine whether the computer has accomplished the task you had in mind. (This is somewhat akin to checking the answer you produced when solving a math problem.)
Essentially the same types of questions are applicable in many other academic subject areas taught in our schools. As it becomes more and more common for students to have good access to Internet-connected computers both at school and at home, we need to restructure our precollege educational system in ways that will make much more effective uses of these rapidly expanding computer capabilities.
What You Can Do
This IAE Blog emphasizes various aspects of sense making, question answering, and problem solving that are of lifelong value. These can be practiced routinely throughout one’s everyday life. Remember that computers are a powerful aid to problem solving, and that their capabilities are being steadily improved.
In your everyday life and as you interact with others, practice the ideas about sense-making stressed in this blog. Think about the information you are sharing, how it makes sense to you, and what sense it will make to your audience. Similarly, when you receive a communication from another source, analyze it in terms of whether it makes sense. Always strive to catch incorrect information and fake news.
References and Resources
Albrecht, R., & Moursund, D. (12/24/2018). Math word problems divorced from reality. IAE-pedia. Retrieved 6/17/2020 from http://iae-pedia.org/Math_word_problems_divorced_from_reality.
Farmer, L. (5/31/2018). Using LibGuide to recognize fake news. IAE Newsletter. Retrieved 6/15/20202 from https://i-a-e.org/newsletters/IAE-Newsletter-2018-234.html.
IAE (2020). Information Age Education main page. IAE-pedia. Retrieved 6/17/2020 from http://iae-pedia.org/Main_Page.
ISTE (2020). We are ISTE. Retrieved 6/16/2020 from https://www.iste.org/.
Math Learning Center (2020). Free math apps. Retrieved 6/17/2020 from https://www.mathlearningcenter.org/resources/apps.
Moursund, D. (6/15/2020). Introduction to ICTing and mathing across the history curriculum. IAE Newsletter. Retrieved 6/15/2020 from https://i-a-e.org/newsletters/IAE-Newsletter-2020-283.html.
Moursund, D. (4/6/2019). Problem solving. IAE-pedia. Retrieved 6/15/2020 from http://iae-pedia.org/Problem_Solving.
Moursund, D. (2016). Word problems in math. IAE-pedia. Retrieved 6/15/2020 from http://iae-pedia.org/Word_Problems_in_Math.
Quora (5/23/2020). Retrieved 5/23/2020 from https://www.quora.com/You-walk-into-a-room-and-see-a-bed-with-2-cats-4-dogs-and-1-cow-on-it-You-see-2-penguins-flying-over-it-How-many-legs-are-on-the-floor/answer/David-Moursund?__nsrc__=4&__snid3__=8384580165.
Quora (n.d.) Home page. Retrieved 6/15/2020 from https://www.quora.com/.