Place Value Knowledge and Equipment

At its core, place value concepts requires students' need to learn multiple representations of the same idea where: 

1. the same digits, 0 to 9 can be used in any of the places 
2. each digit has a different value depending on the place in which they're located on the place value chart
3. our number system have patterns with ten as its base. 

My students lack understanding in each of the aformentioned concepts. This was clearly expressed when  they didn't recognise place value equipment (boards and blocks) when put in front of them, didn't know how to use the equipment, and were unable to answer place value related questions when manipulating place value blocks. 

Clearly the students did not know where to place the digits on the place value chart.

I was surprised that the students knew so little about using the place value equipment, not least because believe in using place value manipulatives as a part of the teaching process - especially for students that fall behind in maths. Cathy Draper, 2006 (in her paper about high school students, It Is All About Connections) expressed the results of inadequate place value understanding among students: "many students.........at these upper levels have never recognized this pattern - and they are good students". In short, students who didn't know our number system has patterns with 10 as its base, couldn't explain why the same digit had different values according to where it sits in the pace value chart. 

If manipulatives do not get used in a classroom, it could be because "teachers need support making decisions regarding manipulative use, including when and how to use manipulatives, to help them and their students think about mathematical ideas more closely" (Puchner, L., Taylor A., O'Donnell, B., Fick, F., 2008, Teacher Learning and Mathematics Maniuplatives: A collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science and Mathematics). 

As teachers, we must ensure students regularly use place value equipment throughout their years of mathematics learning, not just in one or two year levels but at every level during their years at primary school (for example: addition and subtraction in lower levels and decimal numbers in upper levels). Our students must engage in the patterns and representations involved in the process rather than just moving place value blocks around (Early Childhood and Mathematics, Smith, 2009). I like to see place value equipment used at all levels and in every classroom so that students get the benefit of this equipment which will support them to understanding number patterns better.   

Number Knowledge

Hypothesis generation and testing: identifying and systematically testing possible explanations for the issue. This includes developing a rich picture of relevant aspects of your current teaching.

I like to start the year teaching number knowledge, as I feel that this is an area students need to know first and foremost. As a junior teacher, we always begin teaching students how to read numbers, how to write numbers, understand what a number is, rote counting forwards and backwards, then moving on to its value, ordering and the place value.

In the last 5 years our school focused on teaching problem solving and strategies therein. And number knowledge became an aftermath picked up during the problem-solving tasks if students are found to be lacking in it. This method resulted from two years of PLD from a maths facilitator.

Over the years I have progressively felt strongly this was the wrong approach. And I wasn't the only one. Other teachers from the Manaiakalani cluster also said they felt the method was disadvantaging students in their schools as well.

Additionally, teachers at out school felt the same way and wanted to revert back to teaching number knowledge as the first part of their maths programme. Moreover, because we had a number of ESOL students at our school, teaching problem-solving was difficult without teaching appropriate mathematical language first - before even talking about how to solve a mathematical problem.

So last year I made a conscious decision to focus on teaching number knowledge first. In another class, Year 7 students (who were then in year 6) focused on problem solving and strategies last year as per the current school focus. While teaching undoubtedly took place, I believe the students needed more than teaching number knowledge only during problem solving activities. They also need to be taught number knowledge outside the problem solving context. I feel my support for my contention can be found in the PAT results showing poor number knowledge skills for a high percentage of students in Year 7 this year.

If we look at the students' PAT maths, 66% failed in their number knowledge test. The top mark being 5/7 from student A.

Number Knowledge Result

	Total score	Question number and Answer
		1	2	3	4	22	23	25
		E	C	D	C	D	B	C
Student	7
A	5	E	C	D	C	C	D	C
B	2	A	C	D	B	B	C	B
C	2	A	C	E	B	C	D	C
D	1	B	E	E	D	C	C	C
E	4	D	C	D	C	B	C	C
F	2	D	C	C	C	C	E	A

Given the disappointing number knowledge results, I believe I'm making the right decision to teach number knowledge first on my target students this year. I believe not doing this, risks the students missing out on a good grounding in number knowledge during their younger years, which would allow them to compute and solve more complex problems (both in school and in the real word) further down the track.

Having said that, number knowledge cannot be divorced from problem solving activities. Number knowledge still needs to be covered in problem solving activities - because problem solving often needs number knowledge for resolution. But this doesn't mean that problem solving/number knowledge coexistence is synonymous, "because while number sense is inherent in problem solving, many problems are solved without recourse to number sense." (Hiebert et al., 1997, quoted in the academic work, The Relationship between the Number Sense and Problem Solving Abilities of Year 7 Students by Jenny Lounge and Jack Bana). So it's very important for teachers to see number knowledge not as a topic on its own, but also as a prerequisite to problem solving activities - even if it's not always necessary to resolve a problem.

Hence, just like basic facts is the foundation to all mathematical concepts and understanding (see ARBs Basic Facts Concept Map), number knowledge is the first step to the bigger maths picture. In a sense, my approach represents a test on whether teaching number knowledge makes a statistical difference to my students' understanding of mathematical concepts when they apply their number knowledge to addition, subtraction, multiplication, division, fractions, decimals and of course to other strands, namely geometry, algebra, measurement and statistics.

My decision to teach number knowledge before problem solving also comes with the backing of research. Louange and Bana, 2010, carried out a series of assessment to discover any significance in the relationship between number knowledge and problem solving. Without going into the data analysis, their year 7 student interviews confirmed the data results. As one students said, "I don't think that I did not understand what I read. I understand all these words, but there are calculations to be made, but I don't know which calculation to do. I don't always understand what to do with the numbers". The research showed that students sometimes couldn't resolve mathematical problems even though they could comprehend what the problem was asking.

Another student said, "since most problems require number sense, students with such ability have a great advantage over those with poor or no number sense when it comes to successfully solving a problem." Louange and Bana continued, "all three teachers and the majority (70 percent) of students believed that lack of number sense is a probable major cause of poor performance in solving mathematics problems. Clearly, the link between number sense and problem solving is very significant." In other words, successful problem solving relies more heavily on number knowledge, than it does on knowing what the problem asks to resolve.

My experience teaching number knowledge to my target students so far reflects the potential for similar outcomes in problem solving activities. For example, several lessons into my programme, I discovered my students could not expand numbers once the number got past 10,000, nor could they compact an expanded number. Quite apart from expanding numbers to show some understanding of place value, the value of each digit, and the base 10 number system, they also need to understand the 'expanding' concept in later school years to solve, for example, quadratic equations. Insufficient place value knowledge has also reared its head, though this topic will be discussed in the next blog.

PAT Maths

Profiling: understanding the nature of the students' learning strengths and needs in detail. 

We're lucky in NZ that our PAT maths have been designed with appropriate expertise and "helps teachers determine their students’ levels of achievement in the knowledge, skills and understanding of mathematics in the New Zealand curriculum." (NZCER Rangahau Maautranga o Aotearoa)

Our NZ PATs prevent biases that have been found in poorly designed multiple choice questions, which have been caused by a lack of clarity through unclear wording, using poorly designed options that telegraph the answers to students, testing recall rather than accumulated knowledge, and using questions that inadequately measure skill attainment etc. 

	Stanine
Student A	5
Student B	3
Student C	3
Student D	4
Student E	4
Student F	3

Given the expertise, it's clear the PAT structure cannot be blamed for my students' disappointing results. That makes it doubly important to analyse the answers (to come at a later date) to guide my teaching and programme development to strengthen their weaknesses and close the gaps that exist in their learning. 

PAT testing is our school-wide assessment and as teachers we rely on multiple assessments other than PATs (observation, bookwork, individual conferencing, group and class discussions) to decide our overall teacher judgement of students progress. Notwithstanding the different forms of assessment, and given students must sit PATs, it's important to teach my students how to navigate the multiple-choice format.

I believe it's also important for teachers to be aware of both the advantages and disadvantages of multiple choice formats to make best use of any analysis work of test results, and to more concisely define their levels of trust in the PAT structure. Simkin & Kuechler, 2005 (quoted in Multiple-choice questions: Tips for optimsing assessment in-seat and online, a research paper by Xiaomeng Xu, Samantha Tuby, Sierra Dawn Kaur) mentions some advantages of multiple choice tests:

1. it allows "test-givers to ask a greater number of questions on a broader set" of topics in a shorter amount of space and time, 
2. it makes the "administering and grading the exam simpler" with students having to choose only one of the prerecorded answers and the computer having to mark only what is correct,
3. it reduces the "subjectivity / inconsistency / human errors in scoring" when "machine graded" thus guaranteeing more accurate results,
4. the research also quotes Marsh, Roediger, Bjork & Bjork, 2007 who state multiple choice tests have "been shown to positively enhance retention of the material that is tested (a testing effect) and to boost performance on later test."

The research also mentions multiple choice format disadvantages such as:

1. students can "choose answers based on the process of elimination" which could point to them not having the knowledge ingrained,
2. because each question has a pre-written, pre-determined correct answer, it can make it "difficult to pinpoint a student's true knowledge,"
3. It also quotes Ozuru, Briner, Kirbe & MCNamarra, 2013 who point out that students may have the skill to perform an activity, but lack comprehension skills to understand what has been asked,
4. One other thought: multiple choice questions gives students the opportunity to guess, and if they guess right, they're given credit for something they don't know.

Taking these advantages and disadvantages together serves to enhance my personal perspective: I find the PATs very useful because they have been proven by the test of time; machine marking means I'm guaranteed accurate results; they guide me on the individual's/group's strengths and weaknesses; I can carry out the next learning steps with more confidence; the PAT Maths cover all the maths strands in one test and gives me an overall view of each students' progress. Finally, as an aside, it helps students develop the discipline to sit and complete formal tests written in a formal way before they get to high school. In short, preparing our students to cope with future exams gives them a valuable skill they can fall back on with every exam they take. 

GloSS Data

Profiling: understanding the nature of the students' learning strengths and needs in detail.

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.

	GloSS T2 2020
	addition/ subtraction.	multiplication /division	proportional /ratio
A	E6	E6	E5
B	6	E6	E5
C	E5	E5	4
D	6	E6	2
E	5*	E6	E5
F	6	E5	5

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.