I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.
To the parent toying with the idea of actively participating in their child’s mathematical studies, the adage ‘no good deed goes unpunished!’ might come to mind that disabuses them of the notion.
However, parental interaction coupled with an understanding of the rudiments of mathematics knowledge can provide dividends.
Focus on the recommendations outlined below and be surprised how significantly your contributions can enhance your children’s mathematical studies.
Rote learning is cool
Who said repetition is boring?
Rote learning may come across as tedious, but it is often the only way to consolidate factual knowledge. It is common practice for textbooks to provide exercises with repetitive style questions that focus on a specific task.
There is no better way, for example, to master multiplication tables other than by sheer repetition. Traditionally, teachers recommend that students learn their times tables up to at least 12 x 12. This ostensibly means learning 144 pieces of multiplication, starting with 1 x 1 =1 and going up to 12 x 12 = 144. But, in reality, there are only 72 different multiplication answers. This is because 2 x 3 produces the same answer as 3 x 2, 7 x 6 gives the same result as 6 x 7 and so on. (In mathematics, we refer to this as the commutative property under multiplication).
Firstly, for young children, recite the ‘1 x ’ tables out loud.
1 x 1 =1 , 1 x 2 = 2, all the way to 1 x 12 = 12.
It won’t take long to appreciate that multiplication by 1 produces the same answer as the number being multiplied. To demonstrate the commutative property, read out loud ‘1 x 1 =1’ and ‘2 x 1 = 2’ up to 12 x 12 = 144. Another observation will be that answers comprise, in ascending order, whole numbers from 1 to 12.
After mastery of multiplication by 1 is established, other combinations are introduced.
Along the way, ask for any other properties which present themselves. Prompt with something like, “A square number is obtained by multiplying two numbers which are the same. How many square numbers are there up to 12 x 12 ?”
One appropriate response will be, “There are 12 square numbers, namely 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144”.
Of course, older children require familiarity with well-known formulae such as Pythagoras’ theorem and a range of rules to find areas of 2-dimensional shapes. Although each formula can be derived algebraically, it is easier for the student to commit formulae to memory using mnemonics. For example, SOHCAHTOA is often used to remember the relationships between two sides of a right-angled triangle and the angle between them.
SOH: Sine = Opposite/Hypotenuse
CAH: Cos = Adjacent/Hypotenuse
TOA: Tan = Opposite/Adjacent.
Parents can assist in the creation of new or alternative mnemonics, such as CpiD (pronounced Seepid). This prompts one to recall that the circumference, C, of a circle is C=pi x D).
Word glossary and problem interpretation
Understanding mathematics jargon is important.
Students of mathematics encounter words and definitions exclusive to the study. For instance, 'vector', 'discriminant', 'factorising' and 'standard deviation' are commonly used in mathematics but they are not typical topics of conversation around the dinner table. A clear awareness of the meaning of a word is important in understanding the content of a question, because it is often the case that a student can provide an answer if they understand what the question is asking!
For example, Peter knows how to 'multiply the denominator by its conjugate', but he is stumped if the question asks to ‘rationalise the denominator’, even though the two tasks are identical.
As a parent, ask for a description of each key term and invite examples to demonstrate concept acquisition. At the elementary level, this can take the form of direct questioning, with information obtained from the student’s text or notebook.
“Tim, what are prime numbers?” or “Alice, give an example of multiplying two negative numbers.”
For older students, questioning can be open-ended.
“Chien, what methods can you apply to sketch a linear graph using its equation” or “Jonah, what are the main differences between vectors and scalars?”
Additionally, encourage your child to spell glossary terms correctly and provide, where possible, suitable synonyms.
At the junior level, we have “quotient” with “divide”, and at the senior end, “apply the distributive rule” can be paired with “expand brackets”.
You may also want to spend time to assure yourself that the student can differentiate between similar sounding words such as ‘determinant’ and ‘discriminant’. Determinant refers to matrices and discriminant involves quadratic equations.
Other examples include scale/scaler, divisor/dividend, add/addendum, amplitude/altitude/magnitude/latitude, mean/median, iterate/integrate.
Further, offer encouragement to ‘think outside the square’. If one approach does not seem to work, ask for alternative ways, even if the methods have not been formally encountered in the course.
Pride in penmanship
Mathematics is a precise study requiring accuracy.
Presenting clear, unambiguous explanations is a must if students are to perform well in mathematics. Their work should illustrate understanding of the requirements of the question and the appropriate implementation of correct solution methods. This requires knowledge of the range of mathematics symbols and terms appropriate to the level of difficulty of the question. Parents should ensure that this occurs by encouragement, prior to reaching the solution, by requesting verbal explanations of the steps to be followed in a problem-solving process.
As an example, suppose the student’s task is to evaluate the perimeter, in centimetres, of the rectangle shown below.
Some examples of sloppy, inaccurate, unclear or misleading responses might include the following.
80 + 0.05 = 80.05 cm
80 mm = 8 cm 0.05 m = 5 cm 8 + 5 x 2 = 26
P = 0.05 x 100 + 80 ¸ 10 = 13 x 2 = 26 cm
Point out that presenting a solution to a problem is analogous to writing a recipe. The ingredients are the formulae and each instruction must be carried out in the order given. Using incorrect ingredients (formulae) and inaccurate amounts (wrong application of formulae) will not produce the cake (solution).
The right tool for the right job
Choosing the right approach to solve a problem can be problematic!
A study of mathematics incorporates the use of calculators, graph paper, protractors, compasses, rulers, tangrams, trundle wheels, tape measures, weight scales, base ten blocks, dice, pretend money and many more resources and concrete materials. Students must appreciate that each resource serves a specific purpose.
As a parent, ensure at the commencement of a topic that your child has all necessary resources to facilitate task completion. Compromise is usually not a good option.
For example, a ruler can be carefully used to display the graduated axes of a graph, but the use of grid paper will eliminate measurement errors and minimise task time.
By the same token, given enough time and patience, a compass can be creatively used to construct angles, but a protractor is a much better alternative.
Choosing the right resource for the right task also extends to selecting and implementing the right solution method.
When asked to evaluate 6 + 6 + 6 + 6, a junior might perform a sequence of additions.
Thus, 6 + 6 = 12, 12 + 6 =18 and 18 + 6 =24
The enterprising junior might invoke their knowledge of times tables by finding 4 x 6 = 24.
At a more advanced level, ‘mathematical elegance’ comes into play. This essentially means accommodating a task with minimum computational time and using the simplest methods available.
As an example, suppose the aim is to calculate the area of the trapezium shape shown below.
The area can be found by finding the sum of the area of rectangle, R, and the area of triangle T.
Area R = 6 x 8 = 48 square metres
Area T = (2 x 6)/2 = 6 square metres
Total area is R + T = 48 + 6 = 54 square metres
An elegant approach would be to use the formula for the area of a trapezium.
Applying this formula using a = 8, b = 10 and h = 6 gives
This method saves time and provides the answer in a more direct way.
To promote problem-solving skills, the thoughtful parent will regularly ask their child if there are alternative methods to achieve the same results.
To demonstrate this point more clearly, imagine that the task is to find the sum of the first 1000 whole numbers. That is, 1 + 2 + 3 + 4 + 5 + ….. + 1000.
Certainly, a calculator can be laboriously used. Assuming it will take 1 second to input each digit in the calculator, the total time required will be 1000 seconds, or approximately 17 minutes.
The elegant method is to use the formula for the sum of N terms of an arithmetic series.
For 1000 numbers, the value of N to use is 1000. Substitute this value into the formula to obtain
The time needed to enter the computation 1000(1000 + 1)/2 into the calculator is negligible compared to the 17 minutes required if the formula is not used.
When all else fails
So what can be done when an impasse is reached?
If your child is struggling to complete a task and your assistance is limited, recommend that help is sought, even if they feel virtuous and committed by staring at a question for hours, waiting for inspiration. It serves no purpose. Time is better spent finding a solution by communicating with peers, surfing the internet and asking the teacher.
Be a mentor to your children and you will come to be a guiding spirit that nurtures their mathematics education.
Complete the multiple-choice quiz below to gauge your knowledge-level of key mathematical ideas and computations. The content is appropriate up to upper primary level.
The solutions are given at the end of this article.
The number -3 is
A a whole number
B an integer
C a decimal
D a fraction
The number of square millimetres in 2.5 square centimetres is
The fraction 1/7 divided by the fraction 3/14 is
The perimeter of a square is 14 m. Each side of the square has length, in metres, of
Two dice are rolled. The number of times a total of 10 turns up is
The average of the numbers 0, 1, 2, 3, 4 is
If a = 4 and b = 1, then 2a + 3b is
Which statement is always true.
A odd number + odd number = odd number
B even number + even number = odd number
C odd number – odd number = 0
D odd number + even number = odd number
The volume of a rectangular prism (box shape) of length 2 m, width 3 m and height 5 m is
A 30 square metres
B 30 cubic metres
C 10 cubic metres
D 10 square metres
The scale on a map is 1: 5. How long should a shape 40 m in real life long be drawn on paper?
A 8 m
B 35 m
C 5 m
D 20 m
Answers to quiz
1. B 2. C 3. A 4. D 5. B 6. A 7. C 8. D 9. B 10. A