scholarly journals Understanding Mathematical Induction by Writing Analogies

2019 ◽  
Vol 3 (2) ◽  
Author(s):  
Andrew A. Cooper

Mathematical induction has some notoriety as a difficult mathematical proof technique, especially for beginning students. In this note, I describe a writing assignment in which students are asked to develop, describe in detail, critique, defend, and finally extend their own analogies for mathematical induction. By putting the work of explanation into the students' hands, this assignment requires them to engage in detail with the necessary parts of an inductive proof. Students select their subject for the analogy, allowing them to connect abstract mathematics to their lived experiences. The process of peer review helps students recognize and remedy several of the most common errors in writing an inductive proof. All of this takes place in the context of a creative assignment, outside the work of writing formal inductive proofs.

1970 ◽  
Vol 63 (3) ◽  
pp. 249-252
Author(s):  
Richard B. Thompson

The use of mathematical induction seems to be a difficult topic to teach effectively because most applications of induction are either trivial or occur in complicated settings that involve too many ex traneous concepts. Recently, while teaching an elementary course, I gave an assign ment which, to everyone's surprise (my self included), led us to discover one of the basic properties of induction in a par ticularly simple context. We were discussing Pascal's triangle and the binomial co-efficients when I suggested that my stu dents give an inductive proof of the following formula.


1963 ◽  
Vol 56 (8) ◽  
pp. 643-644
Author(s):  
Warren E. Shreve

In a presentation of mathematical induction many high school students do not grasp the inductive concept readily. Further, aligned dominoes or toy soldiers do not always put the idea of mathematical induction within their reach. Presented here are two examples which give different methods of making an inductive proof, one directly and the other indirectly.


1999 ◽  
Vol 9 (2) ◽  
pp. 225-245 ◽  
Author(s):  
ANDREW IRELAND ◽  
ALAN BUNDY

Proof by mathematical induction plays a crucial role in reasoning about functional programs. A generalization step often holds the key to discovering an inductive proof. We present a generalization technique which is particularly applicable when reasoning about functional programs involving accumulating parameters. We provide empirical evidence for the success of our technique and show how it is contributing to the ongoing development of a parallelizing compiler for Standard ML.


2019 ◽  
Vol 20 (2) ◽  
pp. 383-398 ◽  
Author(s):  
Jennifer A. Schmidt-McCormack ◽  
Jessyca A. Judge ◽  
Kellie Spahr ◽  
Ellen Yang ◽  
Raymond Pugh ◽  
...  

Acid–base chemistry is a foundational topic that is taught in courses across the chemistry curriculum. Students often have difficulty distinguishing between the different theories of acid–base chemistry—Brønsted–Lowry and Lewis acid–base chemistry—and applying these two definitions correctly in unfamiliar scenarios. To help students learn these definitions and be able to apply them, an acid–base Writing-to-Learn assignment was developed and evaluated. The Writing-to-Learn assignment involved a three-step process where students constructed an initial draft in response to a writing prompt, participated in peer review, and made revisions based on peer review feedback, before submitting a final draft. This process is informed by sociocultural theory applied to writing, which states that student learning of concepts increases through engagement with their peers’ work and receiving peer feedback on their own writing. To test the efficacy of the acid–base writing assignment, an external assessment, comprised of conceptual questions related to acid–base chemistry and students’ confidence when responding to them, was administered in two groups; a treatment group who completed the Writing-to-Learn assignment, and a comparison group who completed a separate assignment. Additionally, students who completed the Writing-to-Learn assignment were interviewed about their experiences. Regression analysis revealed that students in the treatment group had a greater increase in their conceptual understanding and confidence as compared to the students in the comparison group. The results demonstrate the students could successfully write about the Brønsted–Lowry and Lewis acid–base models separately, but were less successful with connecting these two concepts together in their writing. These results demonstrate the efficacy of Writing-to-Learn as an approach for promoting conceptual learning of acid–base chemistry.


2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Cecille DePass

Spirit and Heart: Indigenous People contest the formal and lived curriculaWe welcome proposals to contribute to the special issue.  Understandably, all proposals and work submitted to the co-editors of the CPI, special issue are to be grounded in Indigenous world views, lived experiences and/or ways of knowing.  Academic and community authors, poets and artists who are interested in contributing to this CPI Special Issue, please, submit a proposal in either a single Word or PDF file to any of the CPI Special Issue, Co-Editors by December 15, 2017.   If your proposal is accepted, the completed work is to be submitted for peer-review by March 15, 2018.  Planned publication date: early Fall 2018.


Author(s):  
Siska Firmasari ◽  
Herri Sulaiman

Kemampuan mahasiswa dalam melakukan pembuktian matematis tidak sama bergantung dari kategori kognitifnya. Salah satu metode pembuktian matematika adalah induksi matematika yang memerlukan pemahaman konsep secara sistematis. Tujuan penelitian adalah untuk mengetahui kemampuan pembuktian matematis mahasiswa yang memiliki kategori kognitif tinggi dan rendah menggunakan induksi matematika. Subjek penelitian ini adalah empat orang mahasiswa tingkat tiga Program Studi Pendidikan Matematika dengan klasifikasi dua orang mahasiswa memiliki kemampuan kognitif tinggi dan dua mahasiswa berkemampuan rendah. Instrumen penelitian yang digunakan adalah lembar tes materi induksi matematika dan pedoman wawancara. Penelitian ini merupakan penelitian deskriptif yang mendeskripsikan kemampuan pembuktian matematis mahasiswa dalam menyelesaikan soal terkait induksi matematika disesuaikan dengan kemampuan kognitif tinggi dan rendah. Hasil penelitian menunjukkan bahwa mahasiswa dengan kategori kognitif tinggi mampu menyelesaikan setiap langkah pembuktian secara benar namun belum sistematis, sedangkan yang berkemampuan kognitif rendah tidak memahami alur pembuktian pada langkah induksi, kekeliruan memahami sifat distributif, dan ketidakteraturan menghubungkan setiap langkah pembuktian. Melalui artikel ini, peneliti berharap dapat menganalisis perlakuan yang tepat pada mahasiswa saat mengajar berbagai materi matematika yang menggunakan prasyarat induksi matematika. Kata kunci: pembuktian matematis, induksi matematika, kemampuan kognitif.   ABSTRACT The students’ ability to perform mathematical proof is different depending on their cognitive category. One of mathematical proofing is mathematical induction which requires concepts understanding systematically. The purpose of this research is to know the ability of mathematical proof using mathematical induction of high and low cognitive category students. The subjects of this study are four third graders of Mathematics Education Study Program. Two students have high cognitive ability and the others have low cognitive ability. The mathematical induction material test sheet and interview guideline are used as research instruments. This is a descriptive research which describes the mathematical proof ability of students in solving problems related to mathematical induction adjusted with high and low cognitive ability. The results show that students with high cognitive category are able to complete each step of proof correctly but not systematically. At the same time, the students with low cognitive ability are not understand the proof steps at the induction step, the misunderstood the distributive property, and the irregularity connect the proof steps. The researcher expects to analyze the appropriate treatment to the students while teaching mathematical materials using mathematical induction prerequisites. Keywords: mathematical proof, mathematical induction, cognitive ability.


2019 ◽  
Vol 6 (1) ◽  
pp. 54-68
Author(s):  
Rezky Agung Herutomo

Proofs are the key component in mathematics and mathematics learning. But in reality, there are still many students who make errors when constructing mathematical proofs. Therefore this study aimed to identify common errors when the students are constructing mathematical proofs. The participant  of this study was 51 of 3rd year students of Mathematics Education Department  who enrolled in Real Analysis course in the second semester of the 2017/2018 academic year. The data of the study were obtained by conducting a test consisting of five questions and interview guidelines. The errors identified in this study were (1) proving general statements using specific examples, (2) inappropriate algebraic manipulation in mathematical induction, (3) incorrect reasoning and assumptions in proving with contradictions, and (4) reasoning errors involving natural numbers in mathematical induction. Hence, further study can be developed learning models that promote the conceptual understanding, logical reasoning, and mastery of mathematical proof techniques.


2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Lowell Abrams

One of the typical challenges facing a mathematics student when writing a proof is the need to understand the interplay of details and broader concepts. I describe a multi-step proof-writing assignment used in a mid-level course for mathematics majors that is designed to help with this challenge by forcing students to incrementally increase their engagement with the various conceptual levels of the material at hand.


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