G’day. I am an Australian, and I have been teaching secondary Mathematics in Asia for nearly 20 years. This free website is about learning lower secondary Maths – particularly geometry and graphing – it is for students, teachers and even researchers. Many students say they find Maths hard, (or they “hate “ Maths), but I have found “hands-on” materials provide a solution to this.
The famous Maths (and Second Language and Reading) teacher Caleb Gattegno (1911- 1988) pioneered the use of geo-boards and other similar tools in the Maths classroom. He claimed that they greatly accelerate students learning, and that students retain their knowledge better. He stated he could teach the first six years of Maths in two years. Gattegno focused mainly on the Primary Level, but the purpose of this website is to show that hands-on materials are just as effective at the Lower Secondary Level.
There is one Gattegno quote from the 1980s, which I love :
“ There is no good system of learning, because we are only concerned with one component : that is, the teacher, and what the teacher does, and we give means to the teachers, thinking that what the teacher does will make the student do better, and we have not been able to substantiate this hypothesis. What is required is to ask the question : ‘How could I improve the learning? ’ ”
As a teacher, I have too often found myself in the situation where I ask my students to recall a formula I taught them the previous semester : they have forgotten it !! This begs the question : Did they ever really learn it ? What exactly did the students learn ? The answer might be : to memorize a formula and then to use it in different ways. This may be clever, but can it be said that the students have mathematical knowledge, or understanding ?
I have been teaching Lower Secondary Maths for nearly 20 years, and the most important thing I have discovered is that students learn best in a hands-on way. Hands-on learning is fun, engaging (for the many, not just the few), and students remember what they have learnt. I use three different sets of tools :
- Compass and Straight Edge
- Probability blocks, dice, coins and cards
1. Geo-boards : I began by making a set of geoboards as an experiment, unsure whether students would like them. They did. In fact they liked them so much, I set about teaching as many topics as I could with geo-boards , and this proved to be more topics than I had first imagined. These include not just ‘shape, area and perimeter’ , but ‘fractions and percentages’, ‘linear equations on the Cartesian plane’, ‘transformations on the Cartesian plane’, ‘volumes and surface areas of solids’, as well as numerous ‘geometry theorems’. See my Youtube video Maths with Geoboards for Lower Secondary Mathematics – an Overview
2. Compass and Straight Edge : In centuries past , if any student was taught Mathematics, an essential part was a study of Euclid with a Compass and Straight Edge. This is no longer the case, and one might ask, “What has been the effect of this ?” Well, the modern student has a much weaker grasp of the rules of geometry. Euclid’s proofs, which students can do themselves on a worksheet with the right guidance, require little memorization, and rely mainly on logic and understanding. The benefit of learning Mathematics this way is that it teaches people to be clear and logical : it is good for their ability to think and argue. Thus, the “usefulness” of such study, which some people question, is not just in strengthening students understanding of geometry, but in general cognitive ability. The “self evident” truths of Euclid have inspired many famous men in history, such as Abraham Lincoln.
3. Probability : Known as a tricky area of Mathematics, it can be made much simpler and clearer with the use of blocks, dice, coins and cards, put into the hands of students. Note that not all students are familiar with them, e.g. some students in Asia have never used playing cards, but if they were to sit an international Maths test, they could easily encounter a question on them. These probability exercises are best done in small groups : the students can discuss the problems, and this works well since some probability problems are counter-intuitive.
- They are fun. And fun means near whole class participation. They also overcome a “fear” of Maths, which is quite common.
- Students can learn more easily , because hands-on materials employ more senses than pen and paper or a computer. Also, mistakes can be more easily corrected, especially on geo-boards.
- An essential feature of any good hands-on material is that it allows a problem to be varied, so that the teacher can give the students task after task, because “Practice makes perfect” That variety is possible with all the hand-on materials recommended here.
- Once students have become used to a particular tool, they become confident and learn quickly, and the teacher can go at a greater pace and teach more advanced skills, beyond the curriculum. In the long run, this saves time. It also helps that the students’ work can be so quickly and easily corrected by the teacher in class , especially with geoboards. Also it is not necessary to use materials for a whole class, e.g. geoboards can be used just for the first 15 minutes.
- Because students have done repeated tasks themselves with their hands, their understanding is deeper and clearer and they retain the knowledge. As a teacher, this is perhaps the most telling point of all.
HANDS-ON Vs. COMPUTER-BASED LEARNING
The internet has made it possible for free education to reach people all around the world, which is a great thing, and this website wishes to be part of that. But this does not mean everything is better online. Below I discuss hands-on vs. online for the three sets of tools I use.
For geoboards, there are indeed online geoboards, but they involve no physical manipulation – the learning experience is just not the same. I would say online geoboards are not nearly as much fun, and I am not inspired to experiment with geometry theorems with an online geoboard the same way I am with a real one. For a compass and straight edge, there is a computer program called Geometers Sketchpad, a very good program which simulates compass work, and accurate compass work too. But this is not how a student learns what can and can’t be done with a compass and straight edge. They only learn about that with a compass in their hands. This is how to start.……… then, as the student becomes more masterful, they can progress to the Geometers Sketchpad. For probability, online sites present probability problems which seem real enough, but they are not as engaging as when cards, dice, coins or coloured blocks are actually in the hands of students, and they are asked to solve a problem concerning them.
I said earlier that students no longer study Euclid with a compass and straight edge. Well, I have seen evidence of university students studying his propositions in special courses. But the propositions of Euclid are within the understanding of a teenager, (lower secondary being 13 – 16 years old), and the fact that these are mathematical proofs combined with a “hands-on” activity appeals to many a teenager. Geometry is also strongly linked to other topics in Maths : e.g. Number Theory, which has applications in computing, but I know of no school curriculum which includes Euclid’s propositions and their proofs, except maybe the Chinese one. So my question is – why wait until university to study these ?
In addition, many fairly easy geometry theorems can be studied on a geo-board, e.g. Angle Bisector Theorem, or The Medial Triangle, but these are rarely taught, even though they are well within the understanding of a teenager.
Of course, not all Lower Secondary Maths topics can be taught this way : it is still necessary to teach other curriculum topics like Indices (in the US “Exponents”) , Inequalities and so on. I teach these topics with power points and my own worksheets. Then, at the Upper Secondary level, things change. The emphasis is heavily on the final exam, and there seems to be no time for hands-on activities. It is also true that the complexity of many topics means that hands on materials are no longer suitable e.g. Calculus cannot be taught with geoboards. When I taught Upper Secondary Maths, I spent the first semester teaching problems from the book, and the second semester doing countless questions from past papers, with few hands-on materials in sight. In the Cambridge IGSCE exam, a compass is in fact required, but it is only to do a very basic exercise about Locus. So the Lower Secondary Level presents the last opportunity in the standard curriculum to use hands-on materials. I am in fact advocating that Lower Secondary Maths and Upper Secondary Maths be taught in different ways. And I am quite confident that the strong understanding gained from using hands-on materials at Lower Secondary Level will help students greatly at Upper Secondary Level.
Gattegno, the so called “inventor” of geo-boards, was responsible, almost solely, for the popularity of Cuisenaire rods and geoboards in classrooms during the 1960s and 1970s. Since then, unfortunately, their use has declined. Why ? From experience, I know the answer. You also need a teacher who knows how to teach with these tools, otherwise they sit in the classroom, unused. Geoboards and Cuisenaire rods are really not like Montessori materials, although they are sometimes spoken of that way. Both require a teacher to set the students a series of well thought out tasks, and to keep challenging them with the next one. In a video on Youtube called “Mathematics at your Fingertips”, you can watch Caleb Gattegno teaching. There, you will see a teacher following a quite strict sequence of well thought out exercises. So before a class using hands-on materials, a teacher needs to plan all the tasks, and have their answers as well.
On this website, I try to show how to teach with hands-on materials. I also cover some other topics in Lower Secondary Maths. This website is free and shall remain so. The videos embedded here are posted on Youtube on my channel “Maths with Geoboards”, the slide shows can be downloaded in both English and Thai, and the worksheets are downloadable as PDFs.
If you have any questions, you are welcome to email me at email@example.com