Mathematical maturity:

How can it inform teaching and

learning of mathematics?

Dr Maarten McKubre-Jordens

and Dr Anne Horton

Mathematical Maturity (MM)

A concept that is ubiquitous, but vague.

Sometimes used as informal prerequisite.

Linked to (but distinct from) mastery of

mathematical techniques / performance.

More holistic judgment of a person.

Much ink has been spilled; however…

Not well-defined(!) in the literature.

Aims

Obtain a grounded definition of

“mathematical maturity”; to go beyond

“know it when you see it”.

Establish methods of measuring MM.

Use methods to develop teaching tools.

Three Phases

Literature review

Survey (mathematics educators globally)

Pedagogy development & evaluation

Existing Literature

Much literature mentions MM.

Precious few sources give a definition.

The ones that do: – Are often mutually inconsistent;

– Sometimes “pass the buck”;

– Are full of advice on how to teach for MM, without any productive follow-up.

– Emphasize importance of MM for learners and educators, but do not back up claims with references or research.

– Focus either on math majors or upper division students.

Definitions – sifting the literature

Learn through understanding (not rote)

Abstraction / abstract-from-concrete, or

abstract alone

Generalizing (pattern internalisation / creation)

Identify sloppy proofs, create own proofs

Develop intuition (feeling of correctness)

Transfer to other disciplines

Communicate mathematics

Educating to increase math

maturity

Moursund notes:

– “1. There are no references to or good

examples of math maturity assessment

instruments.

– 2. There are not extensive sets of materials

(lesson plans, examples for use at various

grade levels, etc.) for use by teachers in

integrating more math maturity content

materials into their math teaching.”

Tao’s three stages of math ed.

1. Pre-rigorous

– Informal, intuitive, hand-waving.

– Emphasis on computations.

2. Rigorous

– One is taught to “do maths properly”.

– Focus on theory, not meaning.

3. Post-rigorous

– Return to intuition, now informed by rigorous background.

Recurring themes (folklore)

Abstraction

Proofs

It takes time

Mathematics as language

Questions

Is there anything concretely measurable in the idea of MM?

Is the informal notion of MM in the community relatively consistent? (Anecdotally – yes)

Implicit assumption of positive correlation between MM and math performance. Is it real?

Can a clearer concept of MM be used to enhance math performance?

The survey

Intended participants: mathematicians &

mathematics educators at tertiary level.

Snowball recruiting.

73 responses (42 “in progress”; so 31

recorded).

Focus on practical aspect.

Survey questions

1. Please define "mathematical maturity"

2. How, if at all, can mathematical maturity be measured?

3. How do you recognise mathematical maturity?

5. How do you teach, if at all, for mathematical maturity?

6. How do you know your teaching [for mathematical maturity] was successful?

7. Please give an example question that you would expect a mathematically mature student to be able to answer.

Define “mathematical maturity”

The ability to follow, and produce, intermediate or advanced mathematical arguments. (proofs)

The ability to see through the notation to the ideas, and manipulate the ideas. (abstraction)

A collection of skills that allows a student to read mathematics, listen to mathematics, understand it, talk meaningfully about it and communicate mathematical ideas to others.

Intentional human insight rather than mere logical reasoning.

How can MM be measured?

Some respondents: it can’t.

Problem solving.

Proof construction.

Flexibility / transfer of skills. – For example, when my 4 year old, after refusing to

believe that 8+0=8, was asked to extrapolate from this to 7+0=?, she showed mathematical maturity when she replied "8"

“Routine” exercises.

In conversation / communication.

How is MM recognized?

Some overlap with previous question, but

significant differences arise. Notably, many

who say it is not measurable, give a

response.

Communication.

Confidence.

Ability to critique proof.

How do you teach for MM?

Some respondents don’t(!).

“basic core” of mathematics.

Role-modelling.

Time and practice.

How do you know the teaching (for

MM) was successful?

Much emphasis on proofs and

transferability of skills.

Proofs.

Essay-type questions.

Seeing alternative methods for a problem.

Test scores.

Time – follow-up later.

Example questions for MM

Many “definition-proof” type questions.

After talking about a particular mathematical fact, ask "Why". A mature student will be able to say something.

What is the key difference between linear algebra over the complex numbers and over the reals?

MM is stage-relative (not an absolute scale). – if one of my students at university extrapolated from

8+0=8 to 7+0=8, I would not be impressed; but I was when my 4yo did it.

From here…

Continue data collection; data analysis

Devise means of measuring MM if definition

turns out to be coherent

Devise teaching strategies for MM, implement at

UC.

(post-project) evaluation, integration.

References – selection

Krantz, S. G. (2012). A mathematician comes of age. Washington, D.C.: Mathematical Association of America.

Moursund, David (2005). Improving Math Education in Elementary Schools: A Short Book for Teachers. Eugene, OR: Information Age Education.

Steen, L. (1986). “Developing mathematical maturity.” In Ralston, A., Young, G. S. (Eds.) The Future of College Mathematics. Proceedings of a Conference/Workshop on the First Two Years of College Mathematics. P 99-110.

Tawfeeq, Dant'e A.. The Florida State University, ProQuest Dissertations Publishing, 2003. 3081451. A model of mathematical maturity at the collegiate level among senior mathematics majors: Exploratory study.

Terence Tao’s Blog: https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

Jo Boaler’s YouCubed clip, “Why students in the US need Common Core Math” https://www.youtube.com/watch?v=pOOW0hQgVPQ

Hare, A., & Phillippy, D. (2004). Building mathematical maturity in Calculus: Teaching implicit differentiation through a review of functions. The Mathematics Teacher , 98(1), pp. 6-12.

Croft, T., Duah, F., & Loch, B. (2013). “I’m worried about the correctness”: Undergraduate students as producers of screencasts of mathematical explanations for their peers – lecturer and student perceptions. International Journal of Mathematical Education in Science and Technology, 44(7), pp. 1045-1055.

Stanley, D., & Walukiewicz, J., Cuoco, A., & Goldenberg, E. P. (2004). In-depth mathematical analysis of ordinary high school problems. The Mathematics Teacher, 97(4), 248-255.

Stewart, I. (2006) Letters to a Young Mathematician, New York: Basic Books.