Inductive reasoning is emphasized in many progressive-oriented math classes, as if the inductive-derived hypotheses stand on their own. Neglected in such approaches is the fact that ultimately such conjectures must be proven using deductive reasoning. Typical of the inductive reasoning type of problems with which students are faced are the growing tile problems, such as the following:
Problems like this are based on inductive reasoning unless it is stated that the sequence of numbers so generated is a specific type of sequence such as arithmetic (and therefore linear), geometric, harmonic, and so on.
Even if one is told that the sequence illustrated by the “growing tiles” depicted is arithmetic/linear, the exercise of deriving it is not readily connected to any math topic. It is more like it aligns with one or more of the Standards of Mathematical Practice such as “Look for and make use of structure”. Such problems are given to students as if such pattern-matching activities are believed to provide students with a problem-solving schema which can then transfer to other problems.
In fact, linear sequences such as those associated with “growing tile” problems are best addressed by the point-slope form of linear equations. This topic is addressed in first year algebra courses, and relates directly to the concepts of graphing and linear equations, such as slope and intercepts.
The point slope form of linear equations is given by:
where m is the slope and x1 and y1 represent the respective x and y coordinates of one or more points through which the graph of the linear equation passes. In the growing tile problem above, we see that the sequence of numbers generated by the tiles is 1, 4, 7 and 10. These can be considered the 𝑦 coordinates associated with x coordinates 1, 2, 3 and 4, so that we have coordinate pairs (1,1), (2,4), (3,7) and (4,10). Taking any two of these points, we find the slope of the line to be 3. Since the line passes through all of the points, we can take any point to represent x1 and y1 in the formula. Taking (1,1) for a point, and plugging in to the equation we obtain:
We then solve for 𝑦:
The point-slope procedure directly connects to what students are learning about linearity and graphing of linear equations. It is much more efficient than the inductive find-a-pattern/guess-and-check method.
Finding mathematical patterns is useful and essential in making conjectures. What is equally if not more important, however, is that in mathematics, inductive guesses ultimately must be proven using the rules of deduction—not induction. (The mathematician Georg Polya talks about induction and reasonable guesses in this video. He states at minute 25.24 that we should not believe our guesses, and at minute 25.53 he makes the point that ultimately such guesses need to be proven.)
In the realm of progressivists’ view of math education, growing tile problems are thought to be useful in training students to look for patterns and make conjectures. At this stage in learning, (typically lower grades, 5th through 8th), such problems are more like IQ tests, however. Their contribution to preparing students to “think algebraically” is questionable and in my opinion, time would be better spent in preparing them in fundamentals, as well as learning to solve multi-step problems that generalize to classes of other more complex types.
Fantastic post. As a Primary teacher posts like yours help keep me on the right track. I used to struggle to include work on generalising and pattern identification and description and did so using the inductive process you described above. I did it because I was told children should be doing it but was never satisfied that it served any purpose beyond keeping children busy. I have not done it in a long while, and have instead turned to teaching core maths skills. I have found children taught the key skills are more engaged with maths and believe they can be successful in it. Nevertheless, I have continued to be nervous I am neglecting something by staying away from the above. Writers critical of progressivist education such as yourself have been crucial in helping me improve my teaching, growing my own knowledge and helping me avoid the empty nonsense promulgated by progressivist education activists.