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In my thesis, I have five chapters, each of which includes sections and subsections. Each of the five chapters has an introduction and conclusion.
Should chapter introductions and conclusions be included in the table of contents?
If yes, should they be formatted as a first heading, like that of the APA style?
The overriding response here has to be "consult your supervisor, or any documentation that you university says on how they want things to be formatted".
However, if neither of those gives useful information, I would simply include them if you are including other headings at the same level. So if each chapter has multiple sections of which the introduction and conclusions are examples, I would format them the same as other sections and include all of the sections in the ToC.
Yes, Introduction and Conclusions are core parts that need to be included in a ToC. Note that it is not necessary that the chapters have these titles although in the vast majority of cases they do.
The Introduction sets the perspective for the chapter and the Conclusions summarizes the important conclusions reached in the discussion. Hopefully the Conclusions tie in with the perspective(s) set in the Introduction since they constitute the head and tail of the chapter and the partial conclusions reached therein.
In cap off, if you have a heading within the main part of the thesis it should be in the ToC and this includes Introduction and Conclusions. In the case of a chapter, it may be worth providing a more meaty, descriptive title for the introduction that ties in with the theme of the chapter. This is in my opinion less so concerning conclusions.
