GloSS is a NZ designed mathematics test to measure which strategies students use in mathematical operations (addition, subtraction, multiplication, proportion, ratios). A strategy is the mental process a students uses to resolve the problem given in the interview. The strategies a student uses then points to the stage level that student uses to calculate problems.
Thus teachers have a guide on where to go next when teaching mathematical strategies to students. It's not a perfect test as students with a lower grasp of language may struggle with their oral explanation on how they arrived at their answer. This could disadvantage the teacher who perhaps may unconsciously use a bit of intuition to judge the strategy the student used and whether it was correctly applied. That means small risks may exist with incorrectly attributing a student their stage in mathematical operations. It also disadvantage the student if the wrong stage gets applied.
Nevertheless, GloSS is a useful tool as it's a 'talking test' that steps away from the traditional 'do the test individually in silence' mode. It gives an insight to a students' thinking that more traditional testing methods fail to give. It challenges students to use language to articulate their thinking so others can understand - in itself a skill useful in the real world, both now and in the future.
A final point about GloSS is that it can disadvantage a lot of ESOL students (5/6 my target students) due to limited English spoken in the home because it's a language-based test and students may have clues how to solve a problem but are unable to explain it.
If we look at the Expectation for Number for The New Zealand Curriculum and the Mathematics Standards the target students should be working through Stage 7 Advanced Multiplicative / Early Proportional Part Whole.
Instead, a quick glance of the target students' GloSS test interview 2 as at 29/5/20 shows that all students have not reached the required standard and sit in the stage E5 to stage 6 range for addition/subtraction and multiplication/division. Poorer results are seen in the proportional/ratio where the students' score ranges from stage 2 to early stage 5. This is summarised in the table below. Work needs to be done in taking the target students to the expected stage 7 to develop advanced multiplicative and early proportional strategies as well as the advanced additive.
The poor results in proportions and ratios reflects inadequate mathematical language skills, and a lack of understanding of the mathematical concepts, especially for student D, an ESOL student who just arrived from the islands this year and is in his first year attending an English-speaking school.
The Tasks: Addition and Subtraction
Task 12: Leanna counted 82 penguins on the beach. Later there were only 44. How many penguins had left the beach?
Task 9: Miriama scored 476 points on a video game. Deb scored 123 points on the same game. How many more points did Miriama score than Deb?
Task 6: I have 84 cards. I give 7 cards to my friend. How man cards to I have left?
The Analysis
Students B,D, E and F were able to answer Task 12.
E and F solved the task by rounding and compensation: 82-40=42, then 40-2=38.
B solved the task using the same strategy but her explanation is misleading, "took away 40 then there were 42 left, then added 2 which left me with 38." If we follow this line of calculation the answer is 44. In the real world her explanation would not be at all acceptable; for example, if chemicals were involved there would be serious consequences.
I am inclined to use my intuition that she probably meant to say 'subtract', rather than 'add'. I felt she knew how to solve the problem but her careless use of a mathematical word led to the error in her explanation. So I decided to give her the benefit of doubt and I did not penalise her. This shows the value of teaching students the importance of correct mathematical language use to explain their steps, so this should be part of any maths programme.
D who recently arrived in New Zealand used renaming strategy.
A managed to get as far as Task 9 and solved the problem using part-whole strategy.
C managed to solve Task 6 by using an early numeracy strategy of counting back, hence his stage E5.
*Although E was able to solve Task 12 he could not solve Task 9. He attempted to use his knowledge of place value and mentally added ones, tens and hundreds separately to get his unknown number, 123 + ___ = 276. His answer was 256. When asked how he got the answer he said he took 123 to get to 476 and it came out 256. Then I asked him to tell me what he meant. He said: 1+2, 2+5, 3+6 and that gives you 476. I asked if he was sure and he nodded instantly without thinking about what he had just said. I gave him some wait time to see if he would check his calculation mentally. The only value that has been added correctly is in the tens place, 2+5=7. Again, using my intuition about the student's response and reaction (or rather lack of) when he was given two opportunities to correct his logic and reasoning I scored him at stage E6 (due to a gap in his maths skills set) despite achieving Task 12.
The Tasks: Multiplication and Division
Task 10: A pack of felt pens cost $8. How many packs of felt pens can you buy for $88?
Task 7: You have 30 balls to put into bags. Each bag can hold 5 balls. How many bags do you need?
The Analysis
Students A, B, D and E achieved Task 10.
Students B, D, and E used their multiplication facts to solve Task 10, (8x11=88 and 88/8=11) while A used a combination of multiplicative and additive strategies (8x10=80, 80+8=88).
C and F were not able to solve Task 10 but managed to complete Task 7. F used his fingers to ski count in 5s, while C explained that 5x30=6. I knew what he meant but did he realise what he said and did he also realise that his statement was not reasonable? Again as with Student B (addition/subtraction) and Student E (for multiplication/division), I used my teacher judgement to score him E5. He did use a multiplicative strategy but mixed up his product with the factor. Or alternatively he does not know nor understand the setting out of a multiplication equation. Like B and E, C needs practise in using appropriate mathematical language to explain his strategy. Although his answer was correct his explanation was not.
As an assessor, I am after the correct answer and the strategy used. But I am also looking for correct language used. As teachers, we fail our students if we don't teach them to use mathematical language correctly. It it like teaching English to students and accepting their grammatical mistakes as being ok. It's not ok to accept inappropriate use of maths language.
The Tasks: Proportions and Ratios
Task 11: Which is more money: one-half of $20 or one-quarter of $40?
Task 8: The white piece is one-quarter of a strip. What fraction is the grey piece?
Task 5: These 15 players have to spread out evenly on the court. How many players should be in each third of the court?
The Analysis
F managed to get as far as as Task 11 by using unit fractions from subtraction facts.
A,B, and C completed Task 5. Students A used multiplication fact, E used division fact, and B used a lot of talking to help her solve the problem. Initially she said, "There is nothing because you can't evenly spread 15 people. 15 is not an even number so one has to stay out." As I was about the end the test, she continued talking and then realised how to solve the problem and, said, "If there were thirds then 5 people. I have 5 and 5 and 5." She used additive partitioning.
Student C solved the problem by moving the images with his fingers and was not able to go further in the test.
Lack of understanding of the English language was a problem for D who was not able to complete any tasks.
Given that GloSS focuses on language to show that learners understand how to solve maths problems, it's important that students can talk about maths concepts and tackle maths problems with appropriate language, knowledge and skill sets.
Where to Next
It's generally accepted that children working on a problem together in a group - and using appropriate language (example: what to say and how to say it), appropriate conversational structure (example: established rules on who can speak and when), with appropriate rules of politeness towards each other, tolerance for differing opinions, acceptance of differing abilities etc - get much more out of their learning than if they sit there working individually in a wholly teacher led mathematics activity.
If a class lacks the necessary skills to experiment and succeed in student-led problem solving activities, it's the teacher's responsibility to give students the tools to effectively work together with appropriate mathematical language. This is not to say teacher-led interactions with students is bad. After all, students cannot learn appropriate mathematical language without being taught. But it's equally true that student-student talk improves educational outcomes as well, and to be successful, students need to be taught structure: knowing what to say, when to say it, how to justify their strategy etc. This means first modelling exercises with students until they can use correct mathematical terminology to explain their approach.
Secondly, the class needs to understand how to work together to reach conclusions in student-group-led problem activities. This process was enacted in a classroom that formed part of a study in a research paper titled, 'Teaching Students How to Use Maths Language to Solve Maths Problems', by Neil Mercer and Claire Sams in 2006.
In their research the targeted students showed that, "children with guidance and practice in how to use language for reasoning would enable them to use language more effectively as a tool for working on maths problems together."
Mercer and Sams points out "that improving the quality of children's use of language for reasoning together would improve their individual learning and understanding of mathematics." (p26,2006)
"More precisely, we have shown how the quality of dialogue between teachers and learners, and amongst learners, is of crucial importance if it is to have a significant influence on learning and educational attainment." (p26, 2006)
Mercer and Sams conclude "that the teacher is an important model and guide for pupils’ use of language for reasoning." (p26, 2006).
We do make a difference in the quality of our learner's education and we cannot underestimate our role in supporting them and helping them to improve their learning outcomes.
