Fundamental Issues in Basic Mathematics and Geometry

This article appears for several reasons, the main one being methodological issues. Widespread obscurity in explanatory approaches—replicated across many tutorials and teaching materials—creates significant gaps in pupils’ knowledge base, which in turn leads to a chain of consequences related to the outlined problem.

Existing methodology approaches

As usual. To start our discussion we may have a set of facts, where from the way will have its place to beginning.

Common Approaches

Lets take as examples official curricula, and recommendations of three English-speaking countries, and those are: United States, United Kingdom and Australia.

The United States will be represented by Study.com, and lessons that align with educational standards in the United States, providing clear explanations suitable for various grade levels: Circle Definition for Kids: Explains that a circle is a shape made up of a curved line where all points are an equal distance from the center point. Circle Geometry: Delves into the properties of circles, including definitions and examples of arcs, sectors, and other related concepts.

Australia – Australian Mathematical Sciences Institute (AMSI). AMSI provides detailed teacher modules that align with the Australian Curriculum, offering in-depth explanations of circle geometry: Circle Geometry Module: Defines a circle as the set of all points in a plane that are a fixed distance (the radius) from a fixed point (the center). It also covers related concepts such as radius, diameter, chord, secant, and symmetry.

And, as a dessert meal, our loved Albion is represented by the UK National Curriculum for Mathematics, which provides a structured approach to teaching geometry, including the properties of circles: Key Stage 1 & 2: Pupils are expected to recognise and name common 2D shapes, including circles, and understand their properties. Key Stage 3: Students should be able to identify and apply circle definitions and properties, including center, radius, chord, diameter, circumference, tangent, arc, sector, and segment.

In all of the listed approaches, we can observe an ordering in which the materials are presented: the circle, as a geometry object, is introduced after foundational geometry concepts such as lines, angles, and polygons, particularly at post-primary levels. Across these curricula, the circle is treated not merely as a figure, but as a fundamental ideal geometric object, forming a basic point of reference for mathematical abstraction and reasoning. Let's step aside from the latest assertion, and set ourselves in the chair of the pupil. The theme is an angle, and the teacher tries to explain the angle's main characteristics, including degrees, sides, and all other related subdivisions. Where the point of intersected lines appears in space? What is space? All these and many collateral questions are born in an unprepared mind, creating a salad of obscurity. Finally, from experience, only a few classmates will cope with the disorder and overcome the chaos of unordered images. And where is the point of possibility for placing all the things on respectful shelves, to its proper place?

Now let me turn you back to the commonly recommended skills that a pupil should be equipped with before entering lessons on fundamental geometric figures.

Based on mentioned curricula approaches I summarised skills, pupil should be acquainted with:

Number and Arithmetic Skills

Counting and number recognition: Recognise numbers, order them, and count objects accurately. Basic operations: Addition, subtraction, and simple multiplication/division concepts. Fractions (elementary level): Understanding halves, quarters, simple partitioning of shapes and sets. Tolerance/approximation: Recognising that measurements can have slight variations; simple rounding. Elementary statistics: Reading simple charts, comparing quantities, understanding “more/less.”

Measurement Skills

Length, weight, and volume: Using non-standard and standard units to measure objects. Comparisons: Longer/shorter, heavier/lighter, bigger/smaller. Time and clocks (basic): Understanding hours, minutes, sequencing events.

Spatial and Geometric Awareness

Shape recognition (pre-2D figures): Identify circles, rectangles, squares in the environment. Spatial relations: Concepts like above/below, inside/outside, next to, near/far. Orientation and movement: Understanding turns, rotations, symmetry in a simple form. Elementary geometric tolerance: Recognising approximate equality of lengths/angles in practical tasks.

Pattern, Sorting, and Logical Skills

Pattern recognition: Sequences, repetitions, growing patterns. Sorting and classification: Grouping objects by color, size, or shape. Comparisons and reasoning: Using “same/different,” “more/less,” and basic logical connections.

Pre-geometry preparation

Lines and curves: Tracing, drawing, and identifying straight vs curved lines. Point understanding: Identifying points in space (dots, intersections in simple grids). Basic angles (informally): Recognising “corner,” “bend,” “turn” before formal angle measurement. Simple coordinate thinking: Grids, rows, columns, and simple positional terms.

All these skills mandatory to proceed the geometry concept, and outlined summary insists next:

Before pupils formally learn geometric figures:

They are introduced to fractions, tolerances, elementary statistics, measurement, patterning, spatial reasoning, and logical thinking.

These skills prepare them to handle squares, triangles, and angles without being overwhelmed by abstraction.

Looks like clear, and reasonable position. But... Oh, always and ewrywhere these 'OH...s'!

Stepping Aside From Dogma Can Change the Universe!

Here I propose you imagine a reverse approach to presenting the material, and in a playful manner we will try to go step by step through the curriculum with a modified order of skills the pupil should be taught.

Existing Schema

Angles:

As now, pupil comes to geometry fundamentals with obscurity of several knowledge areas, Let's take the angle measurements. Any kids know that there are corners, three main corner types, and somewhat like acute (0°–90°), right (90°), obtuse (90°–180°), straight (180°), reflex (180°–360°), and complete or full rotation (360°). But... What are the numbers, and where the degrees come from?

Triangle:

In current ordering same way, the triangle basics outlined before the circles comes to horizon of pupil view, with main triangle-related characteristics:

  1. Triangle angle sum: Sum of interior angles of a triangle = 180°
  2. Exterior angle theorem: Exterior angle = sum of opposite interior angles

Analysis of the knowledge a pupil brings to the circle theme, and reasoning about the representation of abstract objects versus the physical sense of mathematical abstractions in the pupil’s awareness.

Empirical understanding of the pupil's way of thinking.

Let us imagine an experiment:

  1. Draw a right triangle on a sheet of A1 paper, using the right angle as the base reference. Let side A be 30 mm and side B be 60 mm.
  2. Now draw a line parallel to the triangle’s diagonal at a distance of 500 mm.
  3. Next, construct a new right triangle, using this new parallel line to define its diagonal.
  4. Based on the side measurements, recalculate all the angles of the larger triangle.

Based on the results of these calculations, we will usually find that the numbers slightly exceed the tolerances of our measuring tools. This outcome can create confusion when compared with the declared postulates. Here lies the point of doubt—and the opportunity for reflection. Let us use this moment in the proper way! Follow me!

Is the game we are playing fair enough, to spending time?

Use Hands, Do Things, And Take Rewards

Let’s try another order for presenting the material to our kids. Angles and triangles, together with polygons, we leave overboard for now.

Do not be afraid of the needles of the compass, and draw the circle, this is only the task we need to start, ah, forgot! First of all point the dot with a pen or pencil, set the compass needle to the dot, and draw the circle now!

Now, we have a circle. Let me note that at this point the pupil unconsciously begins to understand the circle’s center, and we should use this hint to later introduce the coordinates’ zero point. For now, we need to reinforce this understanding by keeping the pupil in the role of an active explorer.

So we have a circle. Time to change a tool: take a line (ruler, whatever), and draw a line just through the central dot. Mark the dots where the line crosses the circle. Done? The terminology of diameter takes place right here! Maybe enough for today? And this time, you are absolutely at the right point!

New lesson, new adventures, go ahead! This time the pupil should repeat the previous lesson’s operations with a compass by themselves, and demonstrate that the material is firmly remembered from before. Now we do practical measurements. We know the diameter and the center of the circle, and it is time to understand the hemi-diameter. Here arises the question, dear colleagues: how can we find the hemi-diameter of the diameter? Kids should come to a collective scientific discussion at this moment; your role is only to encourage all classmates to participate actively. Eventually, someone may note that the compass is the tool that already gives the right answer! Great! The next task is to mark the center of the diameter with this tool in the proper way. Once this step is completed, there is no need to overload the kids with extra tasks. It’s time to draw and consolidate all the material that has been discovered with you.

Do you like to draw? Then you are welcome! Now we know the diameter, and the terminology of radius takes its place here. Based on the fixed compass position detected as a radius, let’s play with this drawing tool (I hint: we will playfully understand what an ideal triangle is). Start by drawing circles: put the needle of the compass on the circle and draw the first circle with the same position and compass distance as the basic circle. Then move the compass needle to the next intersection mark on the base circle, and continue this way until we return to the initial position. We have six beautiful circles around the base circle. We are drawing the Flower! Hooray! Now it’s time to draw lines. Take any cross as the starting point, skip the next cross, and draw a line from the chosen cross to the next. Continue in this manner, with the starting point always being the line crossing the base circle. Oops, we now have three lines, and these form a triangle. This is explanation time. Here we can introduce several concepts: equality and similarity. To demonstrate similarity, repeat the same procedure but extend the compass settings to maximize the angle. Now it’s time to take scissors! Let’s cut out the triangles and compare all the resulting corners of the triangle shapes. All corners are equal, but the figures are not identical! We may hint that the circle has some standard measurements called degrees, and the triangles we achieved are called ideal triangles, each corner measuring approximately 60 degrees.

Oops, we are coming to complex tasks, but the path is overcome only by those who continue to go, so… Time to introduce the concept of degrees. While explaining degrees, we can turn to the history of standardization in measurements, and discuss where meters, feet, inches, centimeters, and millimeters come from. The history of degrees is more legendary than factual; we may tell the fairy tale and explain the commonly cited link to the Babylonian origins of measurement units, but in my opinion, we need to emphasize the ambiguity of the legend.

All right, step by step we approach the bottle experiments. Hey pirates, we are coming for your bottles—tremble and fear us! All pupils have a bottle, and what about the ropes? Indeed, we need thread, scissors, and a bottle (or glass), all cylindrical in shape. Each pupil should make one single rotation with the thread around the bottle. The teacher then comes to each student with a cutter or razor knife and cuts the thread exactly so that the remainder matches the cylinder’s outer circle. Then we place the bottle on the paper and use the bottle’s outer edge as a template to draw the circle. Now we have a thread and a circle. Time to define the circle’s diameter. We should take a compass and set it to exactly the diameter of the circle. There will be two maximal opposite points, and from these points we create a triangle, then build the opposite mirror of the triangle. In this way, we find the center of the circle. This demonstrates to pupils that there are many ways to define the positioning of a circle’s center, which is essential for the circle as a geometric object. Moving ahead, it is intuitively clear that the thread we manipulated is close to the length of the circle’s outer circumference. Using the diameter fixed by the compass, let’s calculate how many times the diameter fits into the circle’s length. Here, a standard linear measuring tool is required. Finally, we will find that even with tolerances, all pupils arrive at very close results! From this, we can assume that the proportion of diameter to circumference always tells us about the circle’s length, and vice versa. Moreover, this proportion is a constant—and this is wonderful! Without even approaching complex tasks, pupils already clearly understand the origin of π, its main value, and its physical meaning.

Here I just want to show you, my dear colleagues, that even a dogmatic ordering of material can be revised, and this revision can lead pupils to a more comfortable and effective learning process. This approach may significantly accelerate the understanding of subsequent material. Of course, I present this approach for discussion, and such discussions are always welcome.