The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish

Abstract

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.

Appendices

Appendix A—A Selection of Mathematics Word Problems (Second-Level Education)

1.1 A Selection of English Questions:

Question 3

5 is a factor of 20.

The factors of 20 are {1, 2, 4, 5, 10, 20}

The factors of 28 are {1, 2, 4, 7, 14, 28}

In these two sets of factors, a number of factors are common to both sets—1, 2, 4. The highest of these, called the Highest Common Factor (or H.C.F. for short) is 4.

Write out the factors of 10 and 15 and hence find the H.C.F. of 10 and 15.

Factors of 10: ______________________

Factors of 15: ______________________

H.C.F. _______

Question 5

12 is a multiple of 3 as we can multiply 3 by 4 to get 12. 12 is also a multiple of 4, of 6 and of 12. 18 is a multiple of 2, of 3, of 6, of 9 and of 18.

List the next 6 multiples of 3 which are greater than 2.

Question 11

In a class of 30 girls, 17 play tennis and 15 play netball. If all the girls play at least one of these games, how many girls play both?

1.2 A Selection of Gaeilge Questions:

Ceist 2:

Scríobh síos na huimhreacha seo leanas ag baint úsáid as uimhireacha:

• (i) Dhá chéad agus caoga ocht __________________________

• (ii) Trí mhíle, ceithre chéad agus seachtó ocht __________________________

• (iii) Ochtó sé __________________________

• (iv) Deich míle, ceithre chéad agus cúig __________________________

Ceist 8

Le linn treimhse de trí uaire thit an teocht i mBaile Átha Cliath ó 6°c go-3°c, agus i Nua Eabhrac thit an teocht ó 18°c go 10°c. Cé acu des na cathracha a bhraith an ladhdú is mó san teocht?

Baile Átha Cliath 

Nua Eabhrac 

Ceist 10

Ag úsáid luibiní {}, liostaigh na heilimintí des na tacair seo leanas:

m.s. An tacar dos na laethanta don tseachtain a thosnaíonn le D {Deardaoin, Domhnaigh}

• A= An tacar dos na slánuimhreacha réidh idir 11 agus 25. ___________________________

• B= An tacar dos na gutaí san teanga Béarla. ___________________________

• C= An tacar dos na iolraí de 5 idir 8 agus 37. ___________________________

• D= An tacar dos na séasúir don bhliain ___________________________

• E= An tacar dos na dathanna i soilse trachta. ___________________________

Appendix B—A Selection of Mathematics Word Problems (Third-Level Education)

2.1 Question 2

For a college assignment you monitored your daily activities for a week. The activities and amount of time spent doing them are represented in the table below.

Type of activity	Time
Watching TV	Over 3 h/day
Studying	4 h
Sport	0.5 h
Socialising	2–6 h/day
Computer	1 h a few days a week

Would you represent this data on a bar graph? Give a reason for your answer.

Ans.

Question 5

Susan wishes to build a fence around a rectangular lawn. The lawn is 50 m long and 30 m wide. One long side of the fence will be made of stone and the other three sides will be made of wood. Stone costs €6 a metre, and wood costs €3 a metre. How much will the fence cost Susan?

Ans. ________________

Question 7

Sarah was preparing to go to America for the duration of the summer holidays. She needed to change some Euros (€) into American Dollars ($). The exchange rate between the Euro and the American dollar was:

$$1\;Euro = 1.21\,American\,dollars$$

• 1. Sarah changed 1500 Euro into American dollars at this exchange rate. How much money in American dollars did Sarah get?

  Ans. ________________

  On returning to Ireland after 3 months, Sarah had $700 left. She changed this back to Euros, noting that the exchange rate had changed to:

  $$1\,Euro = 1.26\,American\,dollars$$

• 2. How much money in Euros did Sarah get?

  Ans. _______________

• 3. During these 3 months the exchange rate had changed from 1.21 to 1.26 American dollars per Euro. Was it in Sarah’s favour that the exchange rate now was 1.26 American dollars instead of 1.21 American dollars, when she changed her American dollars back to Euros? Give an explanation to support your answer.

  Ans. __________________________________________________________________

Question 9

If the length of a square is increased by 10%, and the width of the same square is decreased by 10%, then the area of the square

• A: decreases by 10%

• B: decreases by 1%

• C: is unchanged

• D: increases by 10%

• E: increases by 20%

Question 13

A lighthouse sends out light flashes with a regular fixed pattern. In the diagram below is the pattern of a certain lighthouse. The light flashes alternate with dark periods.

It is a regular pattern. After some time the pattern repeats itself. The time taken by one complete cycle of a pattern, before it starts to repeat, is called the period. When you find the period of a pattern, it is easy to extend the diagram for the next second or minutes or even hours. In the diagram below, make a graph of a possible pattern of light flashes of a lighthouse that sends out light flashes for 30 s/min. The period of this pattern must be equal to 6 s.

Please fill in the missing words in the following questions. Only one word is required in each of the spaces provided.

Question 17

The numbers 0, 1, 2, 3,..... are called whole numbers or _______________. So 75 is an _______________ but 4 1/3 is not an _______________.

Question 18

Any whole number is divisible by itself and 1. If p is a whole number greater than 1, which has only p and 1 as factors, then p is called a _______________ number. 2, 3, 5, 7, 11, 13, 17, 19 and 23 are all _______________. 14 is not a _______________ since it is divisible by 2 and by 7.