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APOS Theory
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<h2 id="mental-structures-and-mechanisms">Mental structures and mechanisms</h2> <p>APOS Theory emphasizes four mental structures: Actions, Processes, Objects, and Schemas and its mental mechanisms: Interiorization, Encapsulation, Coordination, Reversal, among others. </p> <ol><li><i>Actions</i> are specific manipulations or transformations done to already understood <a href="/facts/Mathematical_object/Juhyow9M">mathematical objects</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> Actions are <i>external</i>, meaning that each step of the transformation needs to be performed explicitly and guided by external instructions.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> For example, someone who can only think about <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> by substituting a number into an expression has an action understanding of functions.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li><i>Processes</i> are actions that have been fully internalized. Someone who has internalized an action into a process can do the action without external instruction, can skip steps, and can imagine performing the process in the abstract without specific inputs being present.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The shift from an action understanding to a process understanding is called <i>interiorization</i>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> For example, someone with a process understanding of a function is able to think about a function as taking in unspecified inputs and applying a transformation to them to produce an output, and someone who has a process understanding of an <a href="/facts/N-tuple/BI1HIxL8"><i>n</i>-tuple</a> is able to mentally consider the concept of an <i>n</i>-tuple for unspecified <i>n</i>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li><i>Objects</i> are cognitive objects coming from the ability to conceptualize the totality of a process and understanding how to put additional structure on that totality.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> The shift from a process understanding to an object understanding is called <i>encapsulation</i>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Someone with an object understanding of functions is able to form a set of functions and define additional structure on that set, such as operations or a <a href="/facts/Topological_space/v2ut7CbK">topology</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Encapsulation is considered the most difficult step in APOS-based instruction.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li> <li><i>Schemas</i> are collections of actions, processes, and objects, and other schemas, and the interactions between them, and the ability to understand in what situations it applies.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> For example, for someone who understands the theory of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>, the mental structure of a vector space is a schema.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li></ol> <p>The mental structures are not necessarily gone through in a linear progression.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Mathematics_education/6hXltmmF">Mathematics education</a></li> <li><a href="/facts/Constructivism_(philosophy_of_education)/AGBx90xm">Constructivism (philosophy of education)</a></li> <li><a href="/facts/Reflective_abstraction/2BfVnk1s">Reflective abstraction</a></li> <li><a href="/facts/Jean_Piaget/gQQBtbCU">Jean Piaget</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). <i>APOS Theory</i>. New York: Springer New York. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4614-7966-6">10.1007/978-1-4614-7966-6</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4899-9825-5.</li> <li>Oktaç, Asuman; Trigueros, María; Romo, Avenilde (March 2019). <a href="https://www.jstor.org/stable/26742010">"APOS THEORY: CONNECTING RESEARCH AND TEACHING"</a>. <i>For the Learning of Mathematics</i>. 39 (1): 33–37. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/26742010">26742010</a>.</li> <li>Tall, David (1999). <a href="https://files.eric.ed.gov/fulltext/ED436403.pdf#page=149"><i>Reflections on APOS theory in Elementary and Advanced Mathematical Thinking</i></a> (PDF). International Group for the Psychology of Mathematics Education.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Arnon et al. 2014, p. 19. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Arnon et al. 2014, p. 19. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Arnon et al. 2014, p. 19. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Arnon et al. 2014, p. 21. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Arnon et al. 2014, p. 20. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Arnon et al. 2014, p. 21. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Arnon et al. 2014, p. 21. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Arnon et al. 2014, p. 21. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Arnon et al. 2014, p. 21. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Arnon et al. 2014, p. 22. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Arnon et al. 2014, pp. 24–25. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Arnon et al. 2014, p. 25. - Arnon, Ilana; Cottrill, Jim; Dubinsky, Ed; Oktaç, Asuman; Fuentes, Solange; Trigueros, Maria; Weller, Kirk (2014). APOS Theory. New York: Springer New York. doi:10.1007/978-1-4614-7966-6. ISBN 978-1-4899-9825-5. <a href="https://doi.org/10.1007%2F978-1-4614-7966-6" target="_blank">https://doi.org/10.1007%2F978-1-4614-7966-6</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Oktaç, Trigueros & Romo 2019, p. 1. - Oktaç, Asuman; Trigueros, María; Romo, Avenilde (March 2019). "APOS THEORY: CONNECTING RESEARCH AND TEACHING". For the Learning of Mathematics. 39 (1): 33–37. JSTOR 26742010. <a href="https://www.jstor.org/stable/26742010" target="_blank">https://www.jstor.org/stable/26742010</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>