IMPACT Live!

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

When I first read that NRC report years ago, productive disposition really stuck out to me. It was during that time early in my teaching career where I was realizing that what I perceived to be good teaching practice, was entirely accurate. Around that time, I made a critical shift from mechanics/procedures (something I became very good at as a student) to conceptual understanding and deeper thinking (something that was not the focus when I was a student). Something practical that I like to do with future teachers (and students in other classes) is "problematize" the problems that we solve. Too often, the situations in the textbook problems, despite using real data, are at best "real-adjacent," but more often wholly unrealistic. They exist to promote and motivate certain mathematical ideas. An example from K-8 would be pretty much any word problem that involves a missing addend. (e.g. Chris has a total of 20 toy cars and trucks. 12 of them cars. How many are trucks?) The characters in those word problems must be the most forgetful people or unobservant people on earth if they can't remember or see the thing that they are missing. 😜 Clearly is Chris can count his cars, he could have just counted his trucks and told us how many there are. The same is true in classic algebra problems. Take for example a problem in which we know the percent discount and the new price, but not the original price of an item. Why would we need to know the original price? How did we not know it at the time of purchase? I'm being a bit facetious here, but the point is that these problems don't necessarily promote a productive disposition given their detachment to reality, but critiquing them and rewriting them does. For many of my students (like yours I'm sure), math is a meaningless, skill-oriented, memory-heavy, struggle, despite moments of positivity and enlightenment. I think that promoting disposition over disillusionment is key. If students can leave me class seeing that math is something that is necessary and around us, that's more important than their ability to parrot responses to contrived (not sensible) problems.  So coupled with the mathematics there needs to be reflection on the mathematics and learning/struggles. This is not to say that I have any answers, but I believe that I do have a productive disposition about my students' learning and my teaching. :)

------------------------------
Chris Sabino
Associate Professor
Harold Washington College
Chicago IL
------------------------------

Original Message:
Sent: 11-02-2021 22:19:29
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

Productive disposition is SO important to students in college algebra, precalculus, calculus etc.  Yes, students need to see mathematics as useful and sensible.  I hope that they will also see it as interesting, beautiful, challenging and inspiring.

Self-efficacy is a key.  The old statement rings true; "whether you think you can or think you can't, you are probably right."  Our assessments and course evaluations count numbers of correct answers, but I believe it would be more useful if we measured changes in student attitudes and beliefs.

We want to send our students to future study in STEM fields.  A productive disposition is far more valuable than skills.  What can teachers do to help students develop it?

------------------------------
Robert Cappetta
Florida SouthWestern State College
FL
------------------------------

Original Message:
Sent: 11-03-2021 11:58:54
From: Chris Sabino
Subject: IMPACT in Action - Productive Disposition

When I first read that NRC report years ago, productive disposition really stuck out to me. It was during that time early in my teaching career where I was realizing that what I perceived to be good teaching practice, was entirely accurate. Around that time, I made a critical shift from mechanics/procedures (something I became very good at as a student) to conceptual understanding and deeper thinking (something that was not the focus when I was a student). Something practical that I like to do with future teachers (and students in other classes) is "problematize" the problems that we solve. Too often, the situations in the textbook problems, despite using real data, are at best "real-adjacent," but more often wholly unrealistic. They exist to promote and motivate certain mathematical ideas. An example from K-8 would be pretty much any word problem that involves a missing addend. (e.g. Chris has a total of 20 toy cars and trucks. 12 of them cars. How many are trucks?) The characters in those word problems must be the most forgetful people or unobservant people on earth if they can't remember or see the thing that they are missing. 😜 Clearly is Chris can count his cars, he could have just counted his trucks and told us how many there are. The same is true in classic algebra problems. Take for example a problem in which we know the percent discount and the new price, but not the original price of an item. Why would we need to know the original price? How did we not know it at the time of purchase? I'm being a bit facetious here, but the point is that these problems don't necessarily promote a productive disposition given their detachment to reality, but critiquing them and rewriting them does. For many of my students (like yours I'm sure), math is a meaningless, skill-oriented, memory-heavy, struggle, despite moments of positivity and enlightenment. I think that promoting disposition over disillusionment is key. If students can leave me class seeing that math is something that is necessary and around us, that's more important than their ability to parrot responses to contrived (not sensible) problems.  So coupled with the mathematics there needs to be reflection on the mathematics and learning/struggles. This is not to say that I have any answers, but I believe that I do have a productive disposition about my students' learning and my teaching. :)

------------------------------
Chris Sabino
Associate Professor
Harold Washington College
Chicago IL

Original Message:
Sent: 11-02-2021 22:19:29
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

I love your quote, Bob.  So true!

------------------------------
Linda Blanco
Professor/Math Department Chair
Joliet Junior College
Joliet,IL
ICTM Board, Univ/CC Director
lblanco@jjc.edu
------------------------------

Original Message:
Sent: 11-05-2021 11:37:09
From: Robert Cappetta
Subject: IMPACT in Action - Productive Disposition

Productive disposition is SO important to students in college algebra, precalculus, calculus etc.  Yes, students need to see mathematics as useful and sensible.  I hope that they will also see it as interesting, beautiful, challenging and inspiring.

Self-efficacy is a key.  The old statement rings true; "whether you think you can or think you can't, you are probably right."  Our assessments and course evaluations count numbers of correct answers, but I believe it would be more useful if we measured changes in student attitudes and beliefs.

We want to send our students to future study in STEM fields.  A productive disposition is far more valuable than skills.  What can teachers do to help students develop it?

------------------------------
Robert Cappetta
Florida SouthWestern State College
FL

Original Message:
Sent: 11-03-2021 11:58:54
From: Chris Sabino
Subject: IMPACT in Action - Productive Disposition

When I first read that NRC report years ago, productive disposition really stuck out to me. It was during that time early in my teaching career where I was realizing that what I perceived to be good teaching practice, was entirely accurate. Around that time, I made a critical shift from mechanics/procedures (something I became very good at as a student) to conceptual understanding and deeper thinking (something that was not the focus when I was a student). Something practical that I like to do with future teachers (and students in other classes) is "problematize" the problems that we solve. Too often, the situations in the textbook problems, despite using real data, are at best "real-adjacent," but more often wholly unrealistic. They exist to promote and motivate certain mathematical ideas. An example from K-8 would be pretty much any word problem that involves a missing addend. (e.g. Chris has a total of 20 toy cars and trucks. 12 of them cars. How many are trucks?) The characters in those word problems must be the most forgetful people or unobservant people on earth if they can't remember or see the thing that they are missing. 😜 Clearly is Chris can count his cars, he could have just counted his trucks and told us how many there are. The same is true in classic algebra problems. Take for example a problem in which we know the percent discount and the new price, but not the original price of an item. Why would we need to know the original price? How did we not know it at the time of purchase? I'm being a bit facetious here, but the point is that these problems don't necessarily promote a productive disposition given their detachment to reality, but critiquing them and rewriting them does. For many of my students (like yours I'm sure), math is a meaningless, skill-oriented, memory-heavy, struggle, despite moments of positivity and enlightenment. I think that promoting disposition over disillusionment is key. If students can leave me class seeing that math is something that is necessary and around us, that's more important than their ability to parrot responses to contrived (not sensible) problems.  So coupled with the mathematics there needs to be reflection on the mathematics and learning/struggles. This is not to say that I have any answers, but I believe that I do have a productive disposition about my students' learning and my teaching. :)

------------------------------
Chris Sabino
Associate Professor
Harold Washington College
Chicago IL

Original Message:
Sent: 11-02-2021 22:19:29
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

I think one of the problems of getting students to see how important and omnipresent mathematics is in the world is that it takes a lot of time to set the situation, gather the data, and explain and there's just no time in the curriculum. It becomes more time efficient to focus on the unrealistic artificial type problems, but Chris you are right. Those are boring and sometimes contrived and don't really give the sense of the usefulness of the mathematics.

I would love to use a textbook that focuses on richer problem sets, but we use an OER resource that is notably light in such problems. And honestly, I don't have time to write my own. I have pulled problems from other textbooks, but I still don't feel like the problem sets give the students a rich mathematical experience. This definitely impacts their productive disposition. Does anyone have any suggestions?

------------------------------
Johanna Debrecht
Assistant Professor
Red Rocks CC / Northern VA CC
Lakewood CO
------------------------------

Original Message:
Sent: 11-03-2021 11:58:54
From: Chris Sabino
Subject: IMPACT in Action - Productive Disposition

When I first read that NRC report years ago, productive disposition really stuck out to me. It was during that time early in my teaching career where I was realizing that what I perceived to be good teaching practice, was entirely accurate. Around that time, I made a critical shift from mechanics/procedures (something I became very good at as a student) to conceptual understanding and deeper thinking (something that was not the focus when I was a student). Something practical that I like to do with future teachers (and students in other classes) is "problematize" the problems that we solve. Too often, the situations in the textbook problems, despite using real data, are at best "real-adjacent," but more often wholly unrealistic. They exist to promote and motivate certain mathematical ideas. An example from K-8 would be pretty much any word problem that involves a missing addend. (e.g. Chris has a total of 20 toy cars and trucks. 12 of them cars. How many are trucks?) The characters in those word problems must be the most forgetful people or unobservant people on earth if they can't remember or see the thing that they are missing. 😜 Clearly is Chris can count his cars, he could have just counted his trucks and told us how many there are. The same is true in classic algebra problems. Take for example a problem in which we know the percent discount and the new price, but not the original price of an item. Why would we need to know the original price? How did we not know it at the time of purchase? I'm being a bit facetious here, but the point is that these problems don't necessarily promote a productive disposition given their detachment to reality, but critiquing them and rewriting them does. For many of my students (like yours I'm sure), math is a meaningless, skill-oriented, memory-heavy, struggle, despite moments of positivity and enlightenment. I think that promoting disposition over disillusionment is key. If students can leave me class seeing that math is something that is necessary and around us, that's more important than their ability to parrot responses to contrived (not sensible) problems.  So coupled with the mathematics there needs to be reflection on the mathematics and learning/struggles. This is not to say that I have any answers, but I believe that I do have a productive disposition about my students' learning and my teaching. :)

------------------------------
Chris Sabino
Associate Professor
Harold Washington College
Chicago IL

Original Message:
Sent: 11-02-2021 22:19:29
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------

Hi, Johanna, I agree that using the right problem is key and is often disappointing to see that textbooks are lucking in that regard. Which textbook are you referring to? 

------------------------------
Kyriakos Kypriotakis
Associate Professor, Mathematics
Joliet Junior College
------------------------------

Original Message:
Sent: 11-05-2021 14:14:42
From: Johanna Debrecht
Subject: IMPACT in Action - Productive Disposition

I think one of the problems of getting students to see how important and omnipresent mathematics is in the world is that it takes a lot of time to set the situation, gather the data, and explain and there's just no time in the curriculum. It becomes more time efficient to focus on the unrealistic artificial type problems, but Chris you are right. Those are boring and sometimes contrived and don't really give the sense of the usefulness of the mathematics.

I would love to use a textbook that focuses on richer problem sets, but we use an OER resource that is notably light in such problems. And honestly, I don't have time to write my own. I have pulled problems from other textbooks, but I still don't feel like the problem sets give the students a rich mathematical experience. This definitely impacts their productive disposition. Does anyone have any suggestions?

------------------------------
Johanna Debrecht
Assistant Professor
Red Rocks CC / Northern VA CC
Lakewood CO

Original Message:
Sent: 11-03-2021 11:58:54
From: Chris Sabino
Subject: IMPACT in Action - Productive Disposition

When I first read that NRC report years ago, productive disposition really stuck out to me. It was during that time early in my teaching career where I was realizing that what I perceived to be good teaching practice, was entirely accurate. Around that time, I made a critical shift from mechanics/procedures (something I became very good at as a student) to conceptual understanding and deeper thinking (something that was not the focus when I was a student). Something practical that I like to do with future teachers (and students in other classes) is "problematize" the problems that we solve. Too often, the situations in the textbook problems, despite using real data, are at best "real-adjacent," but more often wholly unrealistic. They exist to promote and motivate certain mathematical ideas. An example from K-8 would be pretty much any word problem that involves a missing addend. (e.g. Chris has a total of 20 toy cars and trucks. 12 of them cars. How many are trucks?) The characters in those word problems must be the most forgetful people or unobservant people on earth if they can't remember or see the thing that they are missing. 😜 Clearly is Chris can count his cars, he could have just counted his trucks and told us how many there are. The same is true in classic algebra problems. Take for example a problem in which we know the percent discount and the new price, but not the original price of an item. Why would we need to know the original price? How did we not know it at the time of purchase? I'm being a bit facetious here, but the point is that these problems don't necessarily promote a productive disposition given their detachment to reality, but critiquing them and rewriting them does. For many of my students (like yours I'm sure), math is a meaningless, skill-oriented, memory-heavy, struggle, despite moments of positivity and enlightenment. I think that promoting disposition over disillusionment is key. If students can leave me class seeing that math is something that is necessary and around us, that's more important than their ability to parrot responses to contrived (not sensible) problems.  So coupled with the mathematics there needs to be reflection on the mathematics and learning/struggles. This is not to say that I have any answers, but I believe that I do have a productive disposition about my students' learning and my teaching. :)

------------------------------
Chris Sabino
Associate Professor
Harold Washington College
Chicago IL

Original Message:
Sent: 11-02-2021 22:19:29
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

Ok I'll start. My son and I were having a meal at Portillo's (an iconic Chicagoland food joint) when he was home from University of Arizona several years ago as a young student in th Eller School of Management. He lamented that in his high school AP calculus class they never really taught him what to do with all those functions. So I got out a napkin, scribbled a graph of a rate function (like millions of dollars per month profits) and drew some Riemannian rectangles, created a sum, did some unit cancelling and voila! There we have the total profits. I then wrote some official math notation, too, using sigma, limits and integral symbols. His response, "I wish they would have told me that." Out of fairness to his fine teachers, my son was a pretty lay back senior…they very likely did tell him that.
So what problems do I like when making a point in a calculus class? I like those discrete ones like in Stewart where there's a table of motorcycle rates over time intervals and total distance is requested. I like related rates problems that cry out for inclusion of units so the unit canceling makes the results sensible. And in business calc, I love a wide variety of word problems like the ones supplied in the Bittenger text we use. I love to make up crazy functions that have average values that can be considered and calculated without calculus (more closely related to things like weighted grading), then show how the calculus does the same thing for functions where that's not possible.
I'm finding it difficult to do this on my phone, so please excuse typos.
Looking forward to your thoughts!

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL

Original Message:
Sent: 11-01-2021 06:44:51
From: Linda Blanco
Subject: IMPACT in Action - Productive Disposition

The IMPACTful Thoughts blog mentions there are 5 strands from the National Research Council (NRC) referenced in the IMPACT document Proficiency chapter. The 5th strand is Productive Disposition:

                The habitual inclination to see mathematics as sensible, useful, and worthwhile,
                coupled with a belief in diligence and one's own efficacy.

How can one apply productive disposition to show proficiency in mathematics at the intensive level?

------------------------------
Linda Blanco
Professor and Department Chair
Joliet Jr College
Joliet IL
------------------------------