This article also appears as Chapter 12 in my book “Math Education in the U.S.” My high school algebra 2 class which I had in the fall of 1964, was notable for a number of things. One was learning how to solve word problems. Another was a theory that most problems we encountered in algebra class could be solved with arithmetic. Yet another was a girl named Fran who I had a crush on.
The problem (forgive the pun) is ‘trickster’ problems. Where the correct answer relies on the turn of one word/phrase. We turn a maths problem into an English literacy problem, and students where English is a Second Language or have low literacy, struggle to access these problems.
Our preference for problem solving are low floor/high ceiling tasks. Eg. How many students can fit in the school hall? What’s the total cost of haircuts in Australia?
I'm not a fan of one-off problems that don't generalize to anything useful such as the low floor/high ceiling task types that you cite. Students will generally lack the skills required to
solve such a problem, skills such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification. (See http://www.ams.org/notices/201310/rnoti-p1340.pdf for more on non-routine problems)
A year or so into teaching I settled on the personal nomenclature of “practices” and “problems”. Practices being direct application of a method recently taught and problems being questions that had a mathematical structure which requires application of that same method but where this is initially not evident from the surface structure of what is presented. That could be a “word problem”, an “authentic problem” or just a devilishly challenging mathematics problem. A problem might also incorporate other previously learned methods. Sometimes I think “practices” and “problems” are simply at ends of a continuum. Other times I recognise that there are a few very specific ways that practices get embedded in contexts to create “problems”. Anyway, I still like these two words and use them with my students.
Many teachers like to talk about the “deep structure” of a mathematics problem. But I don’t like that language. Saying “deep” makes it sound elusive and out of reach for the ordinary mind. But like you say, the mathematical structure is usually sitting right there just below the surface presentation of the problem. Organise the information in the question logically and mathematically, and … POP! The problem collapses into a routine practice. My light bulb moment in coming to love and be good at mathematics in school was realising that once you understood a concept, everything else was just applying it. That seemed to me, at the time, to make mathematics a much less difficult subject to master. As a teacher, I now also recognise that the best path to understanding a concept is explicit instruction with a lot of practice.
I still use the word “exercises” generally without fussing over its semantics. It’s embedded in educational parlance. My undergraduate textbooks all have the same structure and exercise format. The exercises regularly include everything from practices to problems (and proofs of course) but also “explain” type questions. The Cambridge Essential textbooks we use in Australia organise Exercises into parts: Fluency, Problem-solving (usually wordy and in real life contexts), and Reasoning. It’s a bit forced sometimes but generally helpful. The fluency part is the practice. The rest are the problems. Some reasoning/explain questions don’t fit my simple practice–problem dichotomy. Perhaps they could generally come under the “proof” umbrella (logical mathematical reasoning required). Practices, problems, proofs. The alliteration is neat at least!
Anyway that’s enough rambling reflection. Thank you for sharing this!
It always surprises me that math education researchers don’t examine math competitions to investigate this topic. These are filled with problems that challenge the most motivated and educated students.
The putnum and Olympiad being the extremes. But all of the good ones are similar in requiring insightful thought to do well.
I think the answer is these completions are far from the problem most are trying to solve: how to get unskilled unmotivated students to some proficiency. But they are far closer to the imagined outcome they want of students solving novel problems.
I think that they avoid this topic hints at a fundamental lack of understanding of the problem they are trying to solve.
And to do well in such competitions, one must have experience and practice in the type of problems that math reformers hold in disdain. One must start somewhere to get the feel for various types of problems.
An example of this is a problem that I saw in the algebra book mentioned in my post. The problem stated that two people, Alice walks from school to her house in 20 minutes. Her brother Bob takes 30 minutes to walk from home to the school. If they both start out at the same time, how long does it take for them to meet?
One might try to solve this using variations of the distance = rate x time formula. But a familiarity with work problems makes the solution obvious. It is similar to a problem in which two people are working at a job. One person takes 20 minutes to finish it alone; the other 30 minutes. How long to finish the job if both work together? In one minute each person completes 1/20 and 1/30 of the job. In x minutes the entire job is done: x/20 + x/30 = 1. Solving for x, we get x = 12 minutes. The same equation is used to solve the first problem: It takes 12 minutes for Alice and Bob to meet. The same reasoning applies: In 1 minute, they complete 1/20 and 1/30 of the entire distance. In x minutes, their respective distances equal the entire distance.
To have the insight that allows one to see that the distance problem is a variation of a work problems takes experience and practice.
Why I cherish my Dolciani Algebra text: it provides those worked examples for mixture and rate problems!
The problem (forgive the pun) is ‘trickster’ problems. Where the correct answer relies on the turn of one word/phrase. We turn a maths problem into an English literacy problem, and students where English is a Second Language or have low literacy, struggle to access these problems.
Our preference for problem solving are low floor/high ceiling tasks. Eg. How many students can fit in the school hall? What’s the total cost of haircuts in Australia?
I'm not a fan of one-off problems that don't generalize to anything useful such as the low floor/high ceiling task types that you cite. Students will generally lack the skills required to
solve such a problem, skills such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification. (See http://www.ams.org/notices/201310/rnoti-p1340.pdf for more on non-routine problems)
A year or so into teaching I settled on the personal nomenclature of “practices” and “problems”. Practices being direct application of a method recently taught and problems being questions that had a mathematical structure which requires application of that same method but where this is initially not evident from the surface structure of what is presented. That could be a “word problem”, an “authentic problem” or just a devilishly challenging mathematics problem. A problem might also incorporate other previously learned methods. Sometimes I think “practices” and “problems” are simply at ends of a continuum. Other times I recognise that there are a few very specific ways that practices get embedded in contexts to create “problems”. Anyway, I still like these two words and use them with my students.
Many teachers like to talk about the “deep structure” of a mathematics problem. But I don’t like that language. Saying “deep” makes it sound elusive and out of reach for the ordinary mind. But like you say, the mathematical structure is usually sitting right there just below the surface presentation of the problem. Organise the information in the question logically and mathematically, and … POP! The problem collapses into a routine practice. My light bulb moment in coming to love and be good at mathematics in school was realising that once you understood a concept, everything else was just applying it. That seemed to me, at the time, to make mathematics a much less difficult subject to master. As a teacher, I now also recognise that the best path to understanding a concept is explicit instruction with a lot of practice.
I still use the word “exercises” generally without fussing over its semantics. It’s embedded in educational parlance. My undergraduate textbooks all have the same structure and exercise format. The exercises regularly include everything from practices to problems (and proofs of course) but also “explain” type questions. The Cambridge Essential textbooks we use in Australia organise Exercises into parts: Fluency, Problem-solving (usually wordy and in real life contexts), and Reasoning. It’s a bit forced sometimes but generally helpful. The fluency part is the practice. The rest are the problems. Some reasoning/explain questions don’t fit my simple practice–problem dichotomy. Perhaps they could generally come under the “proof” umbrella (logical mathematical reasoning required). Practices, problems, proofs. The alliteration is neat at least!
Anyway that’s enough rambling reflection. Thank you for sharing this!
It always surprises me that math education researchers don’t examine math competitions to investigate this topic. These are filled with problems that challenge the most motivated and educated students.
The putnum and Olympiad being the extremes. But all of the good ones are similar in requiring insightful thought to do well.
I think the answer is these completions are far from the problem most are trying to solve: how to get unskilled unmotivated students to some proficiency. But they are far closer to the imagined outcome they want of students solving novel problems.
I think that they avoid this topic hints at a fundamental lack of understanding of the problem they are trying to solve.
And to do well in such competitions, one must have experience and practice in the type of problems that math reformers hold in disdain. One must start somewhere to get the feel for various types of problems.
An example of this is a problem that I saw in the algebra book mentioned in my post. The problem stated that two people, Alice walks from school to her house in 20 minutes. Her brother Bob takes 30 minutes to walk from home to the school. If they both start out at the same time, how long does it take for them to meet?
One might try to solve this using variations of the distance = rate x time formula. But a familiarity with work problems makes the solution obvious. It is similar to a problem in which two people are working at a job. One person takes 20 minutes to finish it alone; the other 30 minutes. How long to finish the job if both work together? In one minute each person completes 1/20 and 1/30 of the job. In x minutes the entire job is done: x/20 + x/30 = 1. Solving for x, we get x = 12 minutes. The same equation is used to solve the first problem: It takes 12 minutes for Alice and Bob to meet. The same reasoning applies: In 1 minute, they complete 1/20 and 1/30 of the entire distance. In x minutes, their respective distances equal the entire distance.
To have the insight that allows one to see that the distance problem is a variation of a work problems takes experience and practice.
Yes it is like all the writing on deliberate practice 10000 hours and so on doesn’t exist for some people.