There are many factors that influence teaching in a mathematics class. This influence could be a a positive one or a negative one. One of the biggest factors that I have come across personally is the language barrier for learners where English is not their Home Language of the learners in the class. Their capabilities with regards to English may not be at the level of other learners in the class or even at the level in which the teacher is pitching the lesson. This will create a knowledge void for the learner that is not related to the mathematical content. I wanted to examine how teachers explain simple vs complex concepts in Grade 10 multi-lingual classrooms and to do this I used a questionnaire.
The questionnaire had eight concepts which are crucial to the South African Grade 10 mathematics curriculum. Not all of them are introduced for the first time in Grade 10, however a good understanding of them is necessary to be successful in mathematics in Grade 10. The concepts were “Right angle”, “Hypotenuse”, “Sin trig ratio”, “Perpendicular”, “Parallel”, “Rhombus”, “Denominator” and “Asymptote”. These concepts were chosen since they are popular topics that learners struggle with. Some of these concepts are introduced from well before high school, like ”Denominator”, and some are introduced in Grade 10 for the first time, like “Sin trig ratio” and “Asymptote”. There were two columns given per concept. The first column was for a simple (informal) explanation of the concept (that may include manipulatives) and the second column was for a more complex mathematical (formal) explanation of the concept. The definition of a manipulative is “any of various objects or materials that students can touch and move around in order to help them learn mathematical and other concepts”. These questionnaires will be examined, based how much mathematical language is used, to determine if the concepts are really being explained simply or if they are actually more complex that what the teacher believes. In addition, they will be used to see if the answers are actually complex and formally mathematical, or not. Five teachers from the sample said that they did progress the definitions from the start of a section to the end. They will be named KS, JF, RV, DB and JM.
|Right angle||It is the angle that I make with the floor when I stand on it.||It is an angle with measure 90 degrees.|
There were 3 of the 5 teachers (namely KS, JF and RV) that mentioned a real world example or a diagram to be used for the simple explanation. The other two teachers had a very similar description that used degrees for both simple and complex. This showed that there would not be a movement in language in the classes of JM and DB. The word degrees is a scientific term and thus not an everyday term. It is a measure of the size of the angle. It is not immediately obvious what a degree is from the name and thus, the concept would have to be taught.
|Hypotenuse||The side in a triangle, besides the two sides that make up the angle of the corner of a box.||The longest side in a right angled triangle, which would be found opposite the right angle.|
There were 4 out of 5 teachers that used the longest side definition and/or the side opposite the right-angle definition. Only KS explained the line using a door and drawing a line from corner to corner. This was a more simplified version of an explanation that what I would have used and thus a great result for the concept. Every one of the teachers therefore got the complex definition since that is the one that is given most often in all mathematics classes. Thus, there were four teachers that had either similar or identical definitions for both the simple version and the complex version. Thus, there was no movement in language for 80% of the tested teachers.
Sin trig ratio
|Sin trig ratio||In a triangle, where one angle is like the corner of a box, the sin ratio is a fraction where the top of the fraction is the length of the side across the triangle from the angle that is not the box angle, and the bottom of the fraction is the length of the side across the triangle from the box angle.||In a right angled triangle, the sin ratio is defined as the ratio of the side length opposite the angle that is given to the hypotenuse length, i.e. opp/hyp or y/r.|
There were 4 out of 5 teachers that mentioned the simple explanation involved the words opposite and hypotenuse. If the learner does not understand these words (which is possible since the word hypotenuse is a scientific term and opposite is an everyday word but when used in a mathematical context, it needs more clarification), then the definition will not make sense. Only one teacher, RV, mentioned about giving triangles and getting the learners to measure sides and find ratios of the sides to see that there is a common answer. None of the teaching emphasised that there needs to be an angle of interest besides the right angle! Even in the complex explanation column, only 3 of the 5 teachers mentioned that there is an angle involved that would be used to find the opposite side length. In addition to this, not one teacher mentioned anything about the Cartesian plane notation for triangles which involves using y to denote the opposite side, x to denote the adjacent side and r to represent the hypotenuse. Thus, very little movement from informal to formal across the board of responses.
|Perpendicular||When two lines make a T shape or a + shape.||When two lines intersect at 90 degrees to each other.|
Only one teacher, namely JF, mentions the T shape for the simple explanation. All of the other 4 teachers mention the measure of 90 degrees for the simple explanation. Again, this is a problem since if the learner does not understand the measure of degrees, the explanation will be meaningless to them. One teacher, JM, mentioned an example of where perpendicular lines might occur, but this may create a misconception that this is the only place where perpendicular lines occur.
|Parallel||Lines that are always the same distance apart going in the same direction.||Lines with identical gradients.|
When looking at the simple explanation, the majority of explanations involved having lines that were the same distance apart and never meet. One teacher, namely KS, used real life examples of a bar code having parallel lines or taking the floor and roof as parallel lines. The floor and roof example will not always work though, since if the classroom has a roof in the shape of /\, or even /, then the lines will not be parallel. These roofs could occur if the classroom is in a church, maybe, or even a shack. When looking at the complex explanation, there was not one teacher that explained it mentioning that the gradients of the lines are equal. This concept is crucial for analytical geometry!
|Rhombus||A shape with 4 equal sides.||A rhombus is a simple (non-self-intersecting) quadrilateral where all four sides have the same length. It could be called an equilateral quadrilateral as well.|
Only JM had the simple explanation as stated above. DB had the definition but added in extra ideas for the definition that made it more complex, i.e. opposite parallel lines. KS and JF both mentioned that it was a square that is leaning, possibly from being blown by the wind. The concept here is simple, which is good, except that it relies on the learner knowing what a square is and that it has all sides having equal length. If a learner confuses the name square with some other shape like rectangle, then the windblown shape would end up as a parallelogram, instead of a rhombus. As good as JM’s simple explanation was, when writing down the complex definition, he simply wrote down the properties of a rhombus, not the definition of the shape. Many learners, that I have taught, confuse the definition of a shape with the properties of the shape. Even though the answer is not what the answer should have been, there still a movement towards the more formalised language that has been mentioned in all the properties. Words like diagonals and bisect are academic mathematical terms.
|Denominator||The bottom of the fraction.||The expression below the vinculum, i.e. the number of equal sized parts that would be required to make up one whole item.|
Not surprisingly, almost all of the teachers got the simple explanation since it is the most common version of the explanation, even for home language English learners. RV gave the correct simple explanation; however, continued to make it more complex by adding in part of the definition of a rational number. Because she added the extra bit in, she did not see a difference in the simple and complex explanation. This is understandable, since the simple explanation had the complex one in it. DB and JM got close enough to the complex definition, by mentioning it the number of parts that something is divided into. KS and JF still remained in a fairly rudimentary definition space, when looking at the complex definition.
|Asymptote||A line that the graph goes towards, but never touches or passes.||A discontinuity in the graph that arises when a graph tends to infinity with respect to either dependent or independent variable.|
RV wrote that she would show the idea of an asymptote using a hyperbola and getting the learners to check the y-value for x-values that approach the x-value of the vertical asymptote. There is a problem with this method in that the horizontal asymptote is also there. Each of the other four teachers mentioned the same simple definition as above. As for the complex explanation, this was a different scenario. One teacher, being KS, mentioned the fact that the graph is undefined at the asymptote. Every one of the other four teachers still had a definition involving the line that the graph gets close to. The fact that the graph is undefined (or discontinuous) is a crucial part of the conceptual understanding. The fact that these asymptotes also occur due to the graph having x tending to infinity or negative infinity (or even tending to infinity or negative infinity in the case of a hyperbola) was sadly over looked by all the teachers besides JM. The term discontinuity/discontinuous is also outside of the current South African government based curriculum and thus could be left out of explanations.
For some of the teachers, the explanations given were incomplete. This immediately hurts the learner, since they believe that the explanation they are receiving is all they need to deal with and decipher. If they do not know about a missing element, how are they meant to deal with using it in assessments?
As mentioned before, there is a trade-off involved with explaining these concepts. If the concept is explained in a simple informal manner, this may assist the learner to be able to deal with it in an assessment, but it will not help to develop the learner’s mathematical register. If the concept is explained in a complex mathematical manner, then there stands a chance to develop the mathematical register correctly, but the learner may not understand the language used fully and thus, this would hinder their understanding of the concept. The process of explaining and the sequencing of ideas also forms part of what the teacher is teaching. Thus, for the learners that are not English home language learners trying to learn mathematics, there are multiple challenges that the learners bring to the classroom, but the teacher can be equipped to assist every learner in fighting against the challenges and winning!