By Benjamin Braun, University of Kentucky

The phrase “mathematically mature” is frequently used by mathematics faculty to describe students who have achieved a certain combination of technical skills, habits of investigation, persistence, and conceptual understanding. This is often used both with a positive connotation (“she is very mathematically mature”) and to describe struggling students (“he just isn’t all that mathematically mature”). While the idea that students proceed through an intellectual and personal maturation over time is important, the phrase “mathematically mature” itself is both vague and imprecise. As a result, the depth of our conversations about student learning and habits is often not what it could be, and as mathematicians we can do better — we are experts at crafting careful definitions!

In this post, I argue that we should replace this colloquial phrase with more precise and careful definitions of mathematical maturity and proficiency. I also provide suggestions for how departments can use these to have more effective and productive discussions about student learning.

**Three Psychological Domains**

As I’ve written about previously on this blog, a useful oversimplification frames the human psyche as a three-stranded model:

The intellectual, or *cognitive*, domain regards knowledge and understanding of concepts. The behavioral, or *enactive*, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or *affective*, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning, and when we talk about “mathematical maturity”, what we usually mean is that students have high-level functioning across all three of these areas.

As a first version of a better definition of mathematical maturity, we can specify that students who are mathematically mature have highly developed intellectual, behavioral, and emotional functioning with regard to their mathematical work. When we replace our colloquial phrase with this refined three-domain language, then we can clarify more precisely the distinction between students who have good technical skills but give up too easily (i.e. mature intellectually but developing in their behaviors), or who are persistent problem solvers yet are not confident about any of their results (mature behaviorally but developing emotionally), etc.

**The Five-Strand Model of Mathematical Proficiency**

Once we have become more familiar and fluent with using language that distinguishes between the intellectual, behavioral, and emotional domains, it is useful to further specify proficiency within those domains. One means of achieving this can be found in the 2001 National Research Council report *Adding It Up: Helping Children Learn Mathematics*, where a five-strand model of mathematical proficiency was introduced. While this model was motivated by research on student learning at the K-8 level, in my opinion it is an excellent model through at least the first two years of college, if not beyond. In this model, mathematical proficiency is defined through the following five attributes (see Chapter 4 of *Adding It Up* for details).

*conceptual understanding*— comprehension of mathematical concepts, operations, and relations*procedural fluency*— skill in carrying out procedures flexibly, accurately, efficiently, and appropriately*strategic competence*— ability to formulate, represent, and solve mathematical problems*adaptive reasoning*— capacity for logical thought, reflection, explanation, and justification*productive disposition*— habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

The five-strand model and the three psychological domains weave together well. In particular, one can view the first two strands as refinements of the intellectual domain, the third and fourth strands as refinements of the behavioral domain, and the fifth in alignment with the emotional domain.

In my experience teaching students in their first two years of college mathematics, most significant stumbling blocks for students fall clearly within one of these five strands. For example, when students are able to compute a derivative correctly, but are unable to use that information to find the equation of a tangent line, then this student is succeding in strand #2 but struggling with strand #1. As another example, suppose a student is able to do routine computations and is able to explain how formulas are derived, e.g. the quadratic formula from completing the square, but is challenged by multistep modeling problems such as a max/min problem that requires both introducing and solving an appropriate quadratic function. In this case, a reasonable argument exists that the student “knows the math”, i.e. is proficient with strands #1 and #2, but is struggling to develop mastery of the strategies to apply those skills, i.e. strand #3. As a third example, for students who have a negative view of mathematics and their mathematical capabilities, as related to strand #5, it is challenging to develop the persistence and self-efficacy required to do mathematics successfully.

Much like our mathematical conversations benefit from having clear definitions, our conversations about student learning benefit from having clear and agreed-upon language to describe key components of proficiency. The five-strand model provides an excellent starting point for more clear discussions on this topic.

**Mathematical Proficiency for Majors**

For students studying advanced mathematics, whether they be mathematics majors or math minors in math-intensive major programs, the five-strand model is not a sufficient foundation for articulately discussing mathematical proficiency. In this setting, I feel that one of our most useful resources is the 2015 MAA CUPM Curriculum Guide. Specifically, the following two recommendations copied directly from the Overview to the guide provide an articulate description of some advanced behaviors and intellectual knowledge that majors should attain.

*Cognitive Recommendation 1: Students should develop effective thinking and communication skills. *Major programs should include activities designed to promote students’ progress in learning to:

- state problems carefully, articulate assumptions, understand the importance of precise definition, and reason logically to conclusions;
- identify and model essential features of a complex situation, modify models as necessary for tractability, and draw useful conclusions;
- deduce general principles from particular instances;
- use and compare analytical, visual, and numerical perspectives in exploring mathematics;
- assess the correctness of solutions, create and explore examples, carry out mathematical experiments, and devise and test conjectures;
- recognize and make mathematically rigorous arguments;
- read mathematics with understanding;
- communicate mathematical ideas clearly and coherently both verbally and in writing to audiences of varying mathematical sophistication;
- approach mathematical problems with curiosity and creativity and persist in the face of difficulties;
- work creatively and self-sufficiently with mathematics.

* **Content Recommendation 6: Mathematical sciences major programs should present key ideas from complementary points of view: *

- continuous and discrete;
- algebraic and geometric;
- deterministic and stochastic;
- exact and approximate.

At the major level, the 10 items in the CUPM Cognitive Recommendation and the four items in the CUPM Content Recommendation provide a framework that further extends both the three domains and five strand model. The Cognitive Recommendations are primarily focused on the behavioral and emotional domains and on the third through fifth strands. The Content Recommendations further refine the idea of procedural and conceptual understanding in the first two strands by emphasizing that at an advanced level, students need to understand not only the techniques and concepts themselves, but how those techniques and concepts fit together within a broader vision of mathematics.

**Putting These Into Practice**

I will end this article with a few suggestions for how departments or faculty working groups can put these ideas into action.

- Have two or three faculty jointly present these frameworks/definitions of proficiency during a department seminar or colloquium.
- Gather a team of faculty to review the structure and content of a course for first-year students using the three domain and five strand model. Which of these domains/strands are targeted for development by assignments or activities in the course? Are there any that are being unintentionally omitted from the course curriculum or structure?
- Conduct a similar exercise for a major level course or sequence, this time using the language from the MAA Curriculum Guide. Which of these goals are students being explicitly trained toward? If any of these goals are not treated within that particular course, are there other required courses within the major where students are provided the opportunity to develop in that direction?
- Design a short activity/survey for students in a particular class based on this language. Have the activity introduce the language from one of these frameworks, and ask them to identify activities or experiences in their course that they felt helped them develop with regard to those domains/strands/goals. Discuss the results of this activity/survey with a team of faculty or at a department meeting.

It is important to keep in mind that the best way to be more effective in our considerations of student learning is to frame our discussions within clear and precise definitions of mathematical proficiency. For some courses or departments, the three domain model will be sufficient for this, and for others the five strand model or MAA Curriculum Guide goals will be needed. In any event, we need to move beyond overly-vague discussions of “mathematical maturity” and toward a more sophisticated language to discuss student learning.

I love this topic. There is a lot to unpack. Looking forward to next part.

I ran a version of our Transition to Higher Math / Proofs course that used the complex numbers and the related algebra and functions as the theme. This was very valuable for students in that it touched on many areas of mathematics, required some care with definitions and finding meaning, and had lots of inquiry opportunities. There was a true need for rigor in places, but also a place for intuition. Wrote about the course a while back in PRIMUS.