A Primer on Mathematics Education
A short introduction to modern mathematics education theory
Preface
I was assigned with the task of teaching some classes during next winter term 2025, therefore I realised some pedagogical theory would not do me bad, so I learned the basics of the field and now I am sharing what I have learned from the reference that I have put below.
Abstract
This primer offers a concise introduction to modern mathematics education theory, synthesized from foundational pedagogical perspectives to enhance effective teaching practices. Grounded in the author’s preparation for upcoming teaching responsibilities, it explores Radical Constructivism and Social Constructivism as key theories of learning. The text highlights the importance of integrating both individual cognitive processes and social interactions in knowledge construction, advocating for a pluralistic approach that respects ethical principles and cultural backgrounds to maximize teaching efficacy.
The role of symbols and mediation in mathematics education is examined, acknowledging the shift towards diverse media — including digital tools — that extend beyond traditional symbolic representations. The primer outlines developmental cycles in mathematical learning, delineating phases from kinetic interaction in early childhood to post-formal abstract reasoning in adulthood, each necessitating tailored pedagogical strategies.
Problem-solving is identified as a central component across all learning phases, emphasizing the need to guide students through challenging tasks to foster intuitive understanding rather than merely teaching solution techniques. The importance of understanding students’ motivations, objectives, and social contexts is underscored to address issues of inclusivity and engagement, particularly among minorities and underrepresented groups.
Incorporating technology, especially AI and digital tools, is proposed as a means to enhance mathematical intuition, personalize learning experiences, and prepare students for the evolving demands of the 21st century. The author presents a practical six-step teaching methodology that blends constructivist theories with modern technological resources, advocating for collaborative learning and the integration of AI to deepen understanding.
The primer concludes with reflections on the social dimensions of mathematics education and the imperative to create an inclusive field that connects mathematical concepts with real-world applications through modeling and technology.
Reflections on Theories of Learning
There are many educational perspectives of mathematical education, however I would say that the two main theories are the following perspectives:
- Radical Constructivism
- Social Constructivism
In a nutshell, we can sum up these perspectives as the following:
Radical Constructivism
Rooted in the work of Jean Piaget, classical constructivism emphasizes the individual’s role in constructing knowledge.
Piaget proposed that learners actively build their understanding through processes of assimilation and accommodation, integrating new information with existing cognitive structures.
This approach views learning as an internal, personal endeavour where individuals make sense of the world based on their experiences and prior knowledge.
Social Constructivism
In contrast, social constructivism, influenced by Lev Vygotsky, underscores the importance of social interactions and cultural context in the learning process.
Vygotsky introduced the concept of the Zone of Proximal Development (ZPD), which represents the difference between what a learner can do independently and what they can achieve with guidance.
This perspective suggests that learning is inherently a social activity, facilitated through collaboration, dialogue, and cultural tools.
Is pluralism of theories a good thing ?
These perspectives were introduced in the last century and in some way they give a theoretical basis of how humans attain knowledge in general and so in particular in mathematics, whilst they might seem at first contradicting each other, there are actually some voices that try to merge the many theories that exist and create a general theory which in a sense
is what the field of mathematics education is all about, however there are some authors that do make a case that this is not so much a good idea, as matters of individuality and diversity enter the rule, however it is my opinion that such a merge is possible and beneficial as long as we respect some ethical principles, and we are aware of what is the cultural background of our audience so that we can get maximum efficiency.
Symbols and mediation in Mathematics Education
Symbols are a way to encode information and crystal ideas, however they are above everything a tool that is uniquely human. There are some conceptions that mathematics is above all symbols, however this is not so clear given that there are some Platonic views that consider that intellectual ideas have their place in higher domains, and not so much that crystallization of the actual ideas, that is the symbols.
Whilst it is intuitive to think that what matters is the actual symbols, and in practice that is what is going on, nowadays, it is not so simple given
that students can learn about mathematical ideas over many media, and this is what makes it so exciting nowadays given that a mathematical idea can now be expressed over many medium (computers, calculators, digital boards, LaTeX, AI) and so mathematical symbols are no longer the only source of learning mathematics, albeit symbols still play the main role in what we call the formal learning phase which will be described in the next section.
I would like to add up a final point, which is that there is a new medium which is the digital; I would suggest using it to create intuition about
geometrical facts as well as algebraic and discrete aspects of mathematics, computers provide a great a way to generate intuition and understand things in an intuitive way and to get the why of things., as well as improving digital literacy of students in various pieces of software and even programming languages or using AI.
Cycles in mathematical learning
People learn mathematics by phases, this discovery turns out to be quite interesting given that depending on the age there are different techniques that we should apply for people:
- Kinetics ≤ 5 years
- Symbolic — (6–17) years
- Formal — (18–21) years
- Post-Formal ≥ 22 years
Kinetics
This is a phase where students will be mostly interacting with sensations and the real world; They will be developing the first intuition about the real world, by playing with mathematical toys, for example, as well as playing with games that incentivize mathematical thinking.
Symbolic
Here the students gain the capacity to write and understand symbols such as 1, 2 and +, -, this phase is crucial as it is in this phase that the human rationale starts to develop mathematically the ability to express its ideas in a crystal clear way via symbols, however no formal proofs or generalisations are made. This phase is just getting to know how symbols dance with mathematical intuition and the basic rules of logic to manipulate these symbols.
Formal
In this phase students will be for the first time generalising and proving results via symbolic manipulations, this is a crucial step in understanding how mathematics is built and how to express mathematical ideas in crystal clear way, for the first time the student will have access to the power of simple mathematical proof.
Post-Formal
Here the student not only has the ability to make mathematical proofs but to suggest some new routes and think critically about mathematical ideas as well as contributing and generating theories and understanding complex mathematical proofs, this is as good as it gets.
Problem-Solving and heuristics in didactics for the 21st century
In all the phases described above there is a common point which is problem-solving that turns out to be key to mathematical learning according to research, however teaching students how to problem-solve has no evidence of working, so in a sense exposing them to the maximum numbers of reasoning and challenging them in a constructive way is key to learn mathematics, teaching them how to solve things, not so much.
Just giving problems is not the proper way of doing mathematics, making sure the students are guided by humans or artificial intelligence and exposed to theory as they interact with the material proves to be of most importance as this cycle of problems <→ theory guidance <→ explaining to others will prove to be of most importance, as there is strong evidence that this is the way to go.
One last point is that introducing computers in mathematical tasks should be something that is recurrent, most of the time students are not required to prove things at most levels therefore typically computers with CAS or geometrical software will do the job, things like Maple or Mathematica are good for CAS and things like Python or Geogebra/Desmos
or Matlab are good for numerical and graphing functions.
Make sure we understand the motivations and objectives
This will now enter the more social side of teaching, when thinking of mathematics education we do not immediately think of the social side of teaching, however there is strong evidence that shows that minorities do struggle more with mathematics and there is a certain elitism within mathematics, as well as just sometimes an “I do not care” mentality amongst students.
For this to work we have to consider what class do we have upfront and ask key questions and reflect what are their professional and social background as well as their objectives this can cause quite a stir on the approach and type of problems you will be doing, as well as the way you expose the material of a given course.
Politics and Mathematics ?
Everything interacts with politcs, mathematics is not any different as it is a very important skill set in society. We as mathematics educators have to find a way to make the following work:
- Create an Inclusive Field
- Merge Mathematics ideas with mathematical modelling when possible
- Incetivise Mass Personalization systems with AI
Inclusive Field
We should always be aware that mathematics is not a really diverse field, in fact most leaders in the field are still white men and with my personal experience this will still prevail for a while, furthemore scores tend to be lower in minorities, reading mathematical critical theory, or even feminist mathematical education, should be something everyone is acquainted these days, above all make sure you know your class mathematically and socially, these are decisive factors.
Mathematical Modelling
The only natural way to relate mathematics with society is to make it connect with mathematical modelling, there are extensive situations in society where mathematical modelling is present, for example biological systems or encyption or data analysis or climate change or programming, at all levels we can force different society problems in mathematical terms
not only icentivising problem solving but creating awareness to real world problems.
AI in Mathematics education
AI is getting so big each day that we really never know what comes next, however we do know that this system will get onky better and as of the current days they already do mathematics better than most of the human population, as well as approaching PhD students performances recently, so here it is natural to only get these technologies and incorporated i our current system.
There is great skepticism, and fear, on what the role of AI chatbots might be in the future as everyone basically as private tutor for free these days if they know how to use these AI chatbots. Competing against AI is useless, students will just follow the law of least action, as they should, so we should not only try to use AI but make it a tool of learning as in my opinion chatbots are probably the most undervalued technology tool for education these days. In a future blog I might write on how to use chatbots for mathematics learning as knowing how to use itis probably the most important skill one should have in the short-term.
The recipe for teaching mathematics
In a nutshell you should do the following:
1. Motivate definitions of what you are doing via problem solving and examples
2. Give definitions and mathematical results
3. Make them use AI to understand your ideas better, provide them with prompts
4. Make students think individually in problems related to propositions that you want to teach
5. Make them use AI, such as chat GPT, to help them create intuition on the problems, and even solve them
6. Make them show and explain their solutions to others
7. Explain to them how to solve, only if needed.
8. Repeat
I am using the theories of radical construcvism in step 3–4–5 and in the end step 1–2–6–7 in social constructivism in this way we can use the methods of both worlds, of course students <-> professor intercations will count for the social constructivism as well.
Final Remarks
Note that the problems should use digital as well as traditional methods, as we have already talked about these days we have to create citizens as well as mathematicians.
Make the students work together, working on pairs can be a great experience especially after they have gave a think individually on the problems, they can have very productive discussion.
Try to explore problems that explore spacial, algebraic and analytical notions as diferent students have different different skill sets to share with each other.
Bibliography
B. Sriraman and L. English, Eds., Theories of Mathematics Education: Seeking New Frontiers, 1st ed., ser. Advances in Mathematics Education. Berlin, Heidelberg: Springer-Verlag, 2010. doi: 10.1007/978–3–642–00742–2.
