Thinking Process of Secondary Level Students in Constructing Proof by Mathematical Induction in Terms of Their Attitude toward Mathematics

This research aims to describe secondary-level students' thinking processes in terms of their attitude toward mathematics in constructing proof by mathematical induction. This qualitative research involved two students who were selected from 30 students of second grade of senior high school that categorized into two groups, namely the students who have a positive and negative attitude toward mathematics by using Attitude Toward Mathematics Inventory (ATMI) questionnaire. One student from each category was selected to be given a proving test and interviewed. Students’ process of proving test and interview recording were collected and analyzed to identify their thinking process. This research points out that both students who have positive and negative attitudes towards mathematics can recall the information in their memory regarding the characteristics and steps to prove using mathematical induction. However, the student who has negative attitudes towards mathematics tends to experience some difficulties in processing information, especially in the induction step, caused by panic, depression, and insecurity. The student who has a positive attitude towards mathematics also experiences some difficulties in the proving process, namely understanding what to prove and compiling the induction steps. However, she keeps trying and believes that she would be able to solve it. Hence, she can solve the proof completely.

The Expression of Wnt5a Gene Throughout Definitive Endoderm Induction Process in Induced Pluripotent Stem Cells

Background: Induced pluripotent stem cells (iPSCs) have the ability to proliferate indefinitely and differentiate into three germ layers of ectoderm, mesoderm, and endoderm. Definitive induction is the first and the most delicate stage of differentiation of various iPSC-derived organs. It has been found that the Wnt signaling pathway implicates in embryogenesis, organogenesis, and cell communication. Objectives: In the present study, we aimed to investigate the expression pattern of the Wnt5a gene as an indicator of non-canonical Wnt signaling activity during definitive endoderm induction of iPSCs. Methods: Human iPSCs (RSCB0042) were acquired from Royan stem cell bank of Royan Institute (Tehran, Iran). The iPSCs were cultured on a feeder layer of mitomycin-inactivated mouse embryonic fibroblasts (MEF), and iPSC colonies were collected for embryoid body (EB) generation by suspension culture method. Then endoderm induction step was performed using a series of small molecules. The quantitative real-time PCR was used to assess the mRNA expression of wnt5a, Nanog, OCT4, SOX17, and FOXA2 genes. Results: The production of efficient EBs confirmed by a decrease in Nanog and Oct4 gene expression and the success of DE (definite endoderm) induction step was confirmed by a high expression level of DE specific genes, Sox17, and FoxA2. A significant upregulation of Wnt5a in EB samples and a minor decrease at day 4 was observed. However, the differentiation process followed by an incremental fashion in Wnt5a mRNA expression starting from day 4 of differentiation among the samples of days 6 and 8 (DE stage). Conclusions: Our results suggest that Wnt5a is more activated at the later steps of endoderm induction rather than the early steps, which may be due to the stimulation of canonical Wnt signaling. Finding the expression level of Wnt5a could rise insights for developing more efficient differentiation induction protocols.

The Intertwining Lemma

This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in Π‎A corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.

Kemampuan Pembuktian Matematis Mahasiswa Menggunakan Induksi Matematika

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.

Towards Verified Construction for Planar Class of a Qualitative Spatial Representation

PLCA is a framework for qualitative spatial reasoning, using symbolic objects and relationships between them. To show construction of a PLCA expression, the second and the third authors introduced inductive constructions. They also proved that expressions obtained by inductive constructions are planar (planarity) and planar PLCA expressions can be obtained by inductive constructions (realizability). The former one is proven with Coq proof assistant, but the latter is proven with pen-and-paper.We are currently proving the latter with Coq. We locate some oversights in the original inductive constructions and the proof. In this paper, we report these oversights, re-formalization of inductive constructions and modified proofs. We prove planarity and a base case of realizability with Coq, and induction step of realizability with Coq and pen-and-paper proof.

PLANE REAL ALGEBRAIC CURVES OF ODD DEGREE WITH A DEEP NEST

We apply the Murasugi–Tristram inequality to real algebraic curves of odd degree in RP2 with a deep nest, i.e. a nest of the depth k - 1 where 2k + 1 is the degree. For such curves, the ingredients of the Murasugi–Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base case of the induction and Conway's skein relation as the induction step. As an example of applications, we prove that some isotopy types are not realizable by M-curves of degree 9. In Appendix B, we give some generalization of the skein relation for Conway polynomial.

Differing Mechanisms of Expression for Short- and Long-Term Potentiation

Schulz, Paul E. and Jill C. Fitzgibbons. Differing mechanisms of expression for short- and long-term potentiation. J. Neurophysiol. 78: 321–334, 1997. Long-term potentiation (LTP) is a use-dependent form of synaptic plasticity that is of great interest as a cellular mechanism that may contribute to memory storage. It is the sustained phase of population excitatory postsynaptic potential induced by high-frequency stimulation (HFS). HFS can also induce short-term potentiation (STP), a decremental potentiation lasting ∼15 min. It has been unclear whether STP is simply a reversible form of LTP elicited by subthreshold stimuli or whether it is an independently expressed form of synaptic plasticity. We have attempted to clarify the relationship between LTP and STP in the extracellular recording technique in area CA1 of the adult rat hippocampal slice preparation to test four predictions of the hypothesis that LTP and STP are expressed via the same mechanism. First, occluding LTP expression should block STP expression. Saturating LTP under six different conditions, however, did not occlude STP expression. Second, occluding STP expression should occlude LTP expression. The partial or full occlusion of STP by two maneuvers (increasing the stimulus intensity used for HFS or applying 3-isobutyl-1-methylxanthine), however, did not occlude LTP expression. Third, LTP increases and decreases paired-pulse facilitation (PPF), and STP should have the same effect. STP did not change PPF, however. The first three results, then, suggest that STP and LTP are expressed via different mechanisms. Fourth, STP should be maximal near the LTP induction threshold, and then decrease above it. Surprisingly, STP was maximal at or very close to the LTP induction threshold, but it did not decrease above this threshold. This relationship suggests the possibility that STP and LTP share an induction step(s). What is the function of the independently expressed STP? We find that LTP can be induced by two HFSs, each of which is subthreshold for LTP, if the second is given during STP from the first. This suggests that STP can temporarily lower the LTP induction threshold. Three lines of evidence, then, suggest that STP and LTP may be expressed via different mechanisms; however, the proximity of STP saturation to LTP induction suggests that they may share an induction step(s). STP may also have the very important function of temporarily lowering the LTP induction threshold. Finally, these data suggestion caution in interpreting LTP data obtained <20–30 min after HFS, because they may be contaminated by STP, which appears to have different underlying mechanisms.