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 |
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| 1 | 2 | 3 | 4 | 22 | 23 | 25 |
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| E | C | D | C | D | B | C |
Student | 7 |
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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.
