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The following is a transcriptof a teaching experiment, using the Socratic method, with a regular thirdgrade class in a suburban elementary school. I present my perspective andviews on the session, and on the Socratic method as a teaching tool, followingthe transcript. The class was conducted on a Friday afternoon beginningat 1:30, late in May, with about two weeks left in the school year. Thistime was purposely chosen as one of the most difficult times to enticeand hold these children's concentration about a somewhat complex intellectualmatter. The point was to demonstrate the power of the Socratic method forboth teaching and also for getting students involved and excited aboutthe material being taught. There were 22 students in the class. I was toldahead of time by two different teachers (not the classroom teacher) thatonly a couple of students would be able to understand and follow what Iwould be presenting. When the class period ended, I and the classroom teacherbelieved that at least 19 of the 22 students had fully and excitedly participatedand absorbed the entire material. The three other students' eyes were glazedover from the very beginning, and they did not seem to be involved in theclass at all. The students' answers below are in capital letters.

The Concept and Teaching of Place-Value in Math

That is, at or about the time of the destruction of Jerusalem by the Romans; when the Jews, both in their own land, and in the Gentile world, would have the Gospel first preached to them, but would reject it; upon which the apostles and first ministers of the word would turn to the Gentiles, as the Lord commanded them; when he would turn or change his speech and language towards them, and their speech and language towards him would be turned and changed also: for the words may be taken either way; either of God's speech to the Gentiles, which is his Gospel sent unto them; as it was quickly after Christ's resurrection from the dead, and the rejection of it by the Jews; for many hundred years the Lord took no notice of them; winked at the times of their ignorance; sent no prophet to them, nor any message by anyone to instruct them; yea, he spake roughly to them, in a providential way; in the way of his judgments; particularly they raging and imagining vain things against his Messiah, he spake to them in his wrath, and vexed them in his sore displeasure; see but now he alters the tone of his voice, changes his language, and sends his Gospel to them; which is a "language" of love, grace, and mercy; of peace, pardon, righteousness, and salvation; encouraging souls to believe in Christ for those things: and this is a "pure" speech or language; a pure doctrine, fetched out of the sacred Scriptures; free from the dross of error; unmixed, consistent, and all of a piece; and which has a tendency to promote purity of heart, life, and conversation: or, is a "choice speech" (h); as some render it; it speaks of choice things, more valuable than gold and silver, pearls, and precious stones; the doctrines of it being an inestimable treasure, the unsearchable riches of Christ; and this, by the commission of Christ, upon his resurrection from the dead, was ordered to be spoke unto all nations, or this may respect the different language spoken by the converted Gentiles, when the Gospel should come with power to them; who should speak, as all converted persons do, a different language than they spake before; instead of swearing and cursing, lying, filthy, and frothy speaking, now they speak the language of repentance towards God, confessing their sins, and praying for the pardon of them; the language of faith in Christ, first in a more weak and feeble manner, then with more strength and assurance, believing their interest in him, and in the everlasting love of God, and the covenant of grace; the language of love to Christ, his people, truths, and ordinances; a soul abasing, Christ exalting, and free grace magnifying language; the language of praise and gratitude for mercies received, temporal and spiritual; and especially for Christ, and grace and glory by him: they then speak the language of gracious experience to one another; and in the language of the Scriptures, in the taught words of the Holy Ghost; and, in common conversation, their language is pure, and free from that corruption and vitiosity it was before tainted with: this arises from pureness of heart; from a rich experience of the grace of God; from the teachings of the Spirit of God; and which betrays a man, and shows that he has been with Jesus; this is the language of Canaan, , that they may all call upon the name of the Lord; which sometimes takes in the whole worship and service of God; but, since that is later expressed, it rather intends, in particular, prayer to God; for which men are fitted and qualified, by having a pure language turned to them; or through the Gospel coming with power on them; and by virtue of efficacious grace converting them, and causing them to speak differently from what they did before; and then it is their voice is heard in prayer to God; and which is delightful and pleasant to him, and this is the case of "all" such that have this pure language; there is not a prayerless soul among them: it follows, to serve him with one consent; or, "with one shoulder" (i); the allusion is, either to bearers of burdens, that join together in carrying any burden, who put shoulder to shoulder as they carry it; or else to oxen drawing in a yoke, who are yoked together shoulder by shoulder; hence the Septuagint version renders it "under one yoke": in which it is followed by the Syriac and Arabic versions.

It is easy to see children do not understand place-value when they cannotcorrectly add or subtract written numbers using increasingly more difficultproblems than they have been shown and drilled or substantially rehearsed"how" to do (by specific steps; i.e., by algorithm). By increasingly difficult,I mean, for example, going from subtracting or summing relatively smallerquantities to relatively larger ones (with more and more digits), goingto problems that require (call it what you like) regrouping, carrying,borrowing, or trading; going to subtraction problems with zeroes in thenumber from which you are subtracting; to consecutive zeroes in the numberfrom which you are subtracting; and subtracting such problems that areparticularly psychologically difficult in written form, such as "10,101- 9,999". Asking students to (demonstrate how they) solve (the kinds of)problems they have been "taught" and rehearsed on merely tests their attentionand memory, but asking students to (demonstrate how they) solve kinds of problems (that use the concepts and methods you have been demonstrating,but "go just a bit further" from them) helps to show whether they havedeveloped understanding. However, the kinds of problems at the beginningof this endnote do not do that because they have been contrived specificallyto psychologically mislead, or they are constructed accidentally in sucha way as to actually mislead. They go beyond what the students have beenspecifically taught, but do it in a tricky way rather than a merely "logicallynatural" way. I cannot categorize in what ways "going beyond in a trickyway" differs from "going beyond in a 'naturally logical' way" in orderto test for understanding, but the examples should make clear what it isI mean.