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. 

Basic Facts Data

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

Yesterday the students sat their Basics Facts test. This is a school-wide test carried out in Term 2, which assesses for accuracy. The students are given a total of 50 problems, 2 mins/column, a total of 4 mins.

The results are as follows: 

Target Students - Basic Facts 29/5/20
Student	addition	subtraction	multiplication	division	mixed
A	98%	94%	64%	30%	78%
B	94%	98%	62%	36%	90%
C	94%	88%	82%	50%	72%
D	100%	98%	84%	84%	92%
E	96%	94%	94%	84%	98%
F	100%	98%	74%	46%	92%

A quick look at the basic facts test shows that all students are solid on their addition and subtraction facts, except for student C who needs to work on his subtraction.  All students need to know their multiplication and division facts, while at least two need to work on their mixed facts.

A further analysis of their answers shows that I need to cover the concept and definition of zero in  multiplication  as three students (A, C and D) did not understand the purpose of zero - nothing - in multiplication. Student D consistently wrote his answer as 4 x 0 = 4, 5 x 0 = 5, 9 x 0 = 9, etc

Secondly, student D needs guidance in the concept of division and the division sign, as 7 of his divisions were treated as multiplication. Hence 3 / 3 = 9, 6 / 2 = 12, 8 / 2 = 16, etc.

Thirdly in another instance student C and E followed a pattern of operation rather than looking at the detail of the problem. They failed to observe that the operation had changed so they need to learn to notice, recognise and respond to the operation sign whether it is addition, subtraction, multiplication or division.

Fourthly, student C gets the right answer when dealing with zero in a singular list of subtraction basic facts and a second singular list of addition basic facts. But when the basic facts get mixed up with multiplication, division, subtraction and addition in the same list, he consistently gets the wrong answer in subtraction eg. 2 - 0 = 0. Addition was not a problem: when he adds 4 + 0, he gets the right answer.

Fifthly, students A, B, C, and F need to understand that division is repeated subtraction, division and  multiplication are the opposite of each other and that if they use their knowledge of multiplication facts they can easily solve division problems. A corollary of this could be the students don't know that subtraction is the opposite of addition and vice versa.

Numeral formation needs to improve for students B, E and in particular C who formed her digits poorly. It could be that they wrote hurriedly, or it could be that they need practices on how to write numbers. Regardless, it needs to be covered to reduce their error rate.

Lastly, while observing the students during their basic facts test, not one child checked or proof-read their work. In fact student C turned his test paper over, faced down after each test. While student A had given up during his division test and decided he didn't want to do anymore, hence his low score of 30%.

All the above issues need to be addressed and talked about with the students so they become aware of how they can improve on understanding their skills and techniques.

As a teacher of students who have learning needs I am not at all opposed to students knowing their basic facts. To me the issue of learning basic facts is less about what students should know, and more about how basic facts should be taught.

The traditionalist would argue techniques such as memorisation, rote learning, speed tests etc should be used because they have been proven by the test of time.  The modernist points to the importance of number strategies because it teaches students how to manipulate numbers rather than just memorise them.

Professor of Mathematics Education Jo Boeler, comes down firmly on the side of the modernist in her research, 'Fluency Without Fear: Research Evidence on the Best Ways to Learn Maths Facts'. She argues that "mathematics facts are important but the memorisation of maths facts through times table repetition, practice and time testing is unnecessary" and damages student's attitudes towards maths and their long-term understanding of it.

She argues that an adult who was taught their maths facts by traditionalists might say 7 x 8 = 54 when in fact it's 56. If they're unable to correct themselves, it's likely because they lack an understanding of how numbers fit together.

The student taught by a modernist may know that 7 x 7 = 49 and they have to add 7. Or they might know 7 x 10 = 70 and subtract two lots of 7 to arrive at 56. If the student makes an error, they can correct themselves using their knowledge of numbers rather than memorised facts.

The essence of the argument is that students learn their basic facts anyway without memorisation through the process of learning various mathematical strategies. Moreover, because learning number manipulation offers students a deeper understanding of how numbers fit together, it gives them a logical pathway towards higher level thinking.

However, on occasions we have to look inside ourselves because what works as a teaching tool for one student may not work so well for another. So it follows that memorisation and rote learning techniques for learning basic facts should not be completely abandoned. Instant recall of basic facts is a necessary skill to solve number problems especially in PAT tests where speed is required and where time is limited. Moreover as I have already mentioned in my earlier blog, basic facts is the core to the the various conceptual understanding and knowledge in mathematics (27 April) and so basic facts is a requirement.

Admittedly, as a teaching technique for learning basic facts, memorisation focuses primarily on two senses: the ear (as in chanting) and the eyes (as in flash cards and the like). It can be made more successful by adding in the kinesthetic sense (as in writing out the facts by hand in a frequent maths activity).

An even more successful approach uses all senses at once on a device. Susan Koscinski and David Gast claimed in their research, 'Computer Assisted Instruction With Constant Time Delay to Teach Multiplication Facts to Students With Learning Disabilities', that, "results indicate that the computer-assisted instructional program was effective in teaching multiplication facts to students", who have more difficulty in learning basic maths facts than "their other non-handicapped peers". Thus repetitive learning of basic facts has its place, and by extension, it can also benefit those of normal ability who gain more traction using memorisation and rote learning techniques as well.

Hence, one of the teaching tools I will be using to help my learners to increase their basic facts knowledge and their speed of recall will be the use of online interactive programmes.