“It is important that that learners acquire mathematical understanding by constructing their own meaning…”
So how can children make sense of, and then competently use, the language of mathematics? This article looks at ambiguous language and operational language.
The following interaction between a teacher and student was recorded.
T: Can you calculate the volume of this box? S: um .. [pause] .. no [has a puzzled look]
T: Do you know what volume is? S: Yes, it is a button on the remote.
One of the issues with constructing meaning of the language of mathematics is the use of everyday English terms that have different meanings in the mathematics classroom.
Here is a list of just some of the many words that have a different meaning in mathematics. These words are just some of the homographs: rational, mean, power, odd, face, property, common
The operational language can also be confusing. In the early years ‘and ‘is used for operation of addition. In later years the word ‘and’ is used in an operation for multiplication, e.g. ‘What’s the product of 5 and 4?’
When assessing a student’s ability to solve word problems in mathematics it is so important to consider not only the wording of the question but also the order of the questions. Two consecutive questions, similar to these, appeared in a standardised test. The word ‘altogether’ is used for a different operation in each question. What meanings have the students constructed of the words ‘and’ and ‘altogether’?
Budi puts cards into 4 equal piles.
Each pile has 20 cards.
How many cards does Budi have altogether?
Wati collected 68 cans.
Puti collected 109 cans.
How many cans did Wati and Puti collect altogether?
The construction of operational language is so important for problem solving. In some classrooms students can identify words in a problem, referring to displayed visual mathematical vocabulary.
Although language is heavily involved in constructing meaning in mathematics, the use of visual representations and manipulation of concrete materials all support communication and success in the mathematics classroom. The literacy of numeracy is a challenge for all and an additional challenge if English is an additional language. Our role is to support students to construct meaning as part of the stages of learning mathematics.
ACG School Jakarta
School to School is an annual event hosted by Sekolah Ciputra and dedicated to educators who are willing to learn and share their professional learning with colleagues across East Java. This year, it was held on February 25th. Ms. Hestya and I took led a workshop about math misconceptions in the primary years. This area intrigues me, as I believe all Math concepts can be investigated and explained in a simple way and we don’t need to say “this is the procedure, formula, or theory that you need to remember” to our students, which is the way I was taught. If our students understand how math works, rather than memorize formulas, they will love it.
We started the activity by giving a pretest to the workshop participants to identify misconceptions they had. It was surprising to me that no one answered all the questions correctly. Then we followed up with an activity designed to accommodate the needs of the participants and to refine their misconceptions. We discussed and investigated the following topics:
(1) What is a concept, a conception and a misconception?
(2) What forms of misconceptions occur in primary school?
(3) How do teachers respond to student misconceptions?
(4) What techniques are there to eliminate misconceptions?
To refine the participants’ understanding of Math concepts we did a gallery walk. One important thing that we shared is how Math pre-conceptions leads to further misconception. One example is:
- Students get confused with the alligator/Pacman analogy. Is the bigger value eating the smaller one? Is it the value already eaten or about to be eaten? Do I add what it has eaten?
- In helping students make sense of subtraction they are told to always take the smaller number away from the larger number.
4 – 8 = ?
From this workshop, I have learned that effective teachers understand that mistakes and confusion provide powerful learning opportunities. I believe the quote below reminds us that misconceptions hinders inquiry.
“The worst thing about mnemonics is not that they almost always fall apart, they don’t encourage understanding, and never justify anything; it’s that they kill curiosity and creativity – two important character traits that too many math teachers out there disregard.” -Andy Martinson
PYP 6 Teacher and Year Level Coordinator
Multiplication is one of the basic operations in mathematics. Teaching multiplication does not have to be merely rote memorization of isolated number facts.
Although it is important for students to be quick and accurate in computing, it is equally important for them to understand the concept of multiplying. When there is conceptual understanding, students can make connections across contextual real-life situations. This can later on benefit them when faced with other related mathematical applications.
Using various strategies such as grouping and making arrays, skip counting, repeated addition, writing in words, and commutative property will enhance students’ understanding about multiplication.
In doing multiplication by grouping, a specific number of items is repeatedly grouped. One factor states the number of groups. The other factor states the number of items in each group. The product is the total number of items in all the groups. If the items are orderly arranged in rows and columns, then it is called an array.
Example 1: There are 9 groups of stars. Each group has 6 stars. How many stars are there in all?
Another multiplication strategy is skip counting. It is counting while skipping one or more numbers in a pattern.
Example 2: There are 8 apples in a basket. How many apples are there in 9 baskets?
In class, our students felt overwhelmed when they first heard the word multiplication even if some of them were familiar with this operation. What the students were most scared of was that they thought they had to memorize the times tables right away. However, after being introduced them to different multiplication strategies, including creating a number line, they felt relieved. It was NOT as difficult as they thought! The students became more engaged and confident as they got the chance to use their preferred strategies to solve multiplication questions.
Resources like manipulatives and other materials such as playing cards, beads, paper clips, and dice are also readily available as students learn multiplication.
When students understand the basics, they begin to build on relevant concepts and develop strategies that can help them learn more multiplication facts. When students are confident in their abilities in multiplying, they will have a positive outlook towards learning and consequently be successful in their math experiences in the future.
By: Jenina Siauw and Nancy Benedicta
Grade 2 Teachers
BINUS SCHOOL Simprug