When I first started at Timaru Christian School we, as many Christian Schools, had a weekly memory verse. As teachers we would sit together and do our term planning and brainstorm a list of hopefully 10 verses which could be used as memory verses. Fantastic, over the course of eight years each student would have an arsenal 8 x 40, 320 verses which they had 'hidden in their heart'!
Hmmm...some very diligent students learnt the verse well could not only remember the verse on testing day but some even remembered the verse for longer, weeks, months.
HOWEVER, it became very obvious that the verses simply were not being retained long term AND there was little big picture planning for the verses we thought students really needed to know. Verses about sin and Salvation were rightly or wrongly not covered in many of our unit plans, yet were verses we all wanted the students to know. To cater for this I redeveloped the Scripture Memorisation program with three essential elements of success, repetition, levelling and big picture verses, ie verses we really wanted our students to 'take' with them.
Although the programme is very prescriptive the first and last are set aside for a memory verse which is topical, ie one that relates to the current terms unit of study. There are three different levels (stages) to the memory verses, Year 1-4 = Stage 1, Year 5/6 = Stage 2 and Year 7/8 = Stage 3.
In Stage 1 and 2 students memorise a set of 18 verses each semester. This repartition enhances long term leaning and application. In Stage 3 students learn passages of 8 to 12 verses.
Have a look see if you can adjust the verses to be meaningful to your school.
Isn't it so true that we often do not truly appreciate something as a teacher until the tables are turned and we become a learner? Over the past few years we as a school have had quite a focus on the quality of the feedback we give students. We have begun using rubrics, success criteria, WALTs and feedforward. For some of the teachers this has been not been as easy a task as we would have liked, but we are well on our way.
The importance of the whole feedforward debate has become very real to me recently. This year I decided to immerse myself into the world of post graduate studies and am now two thirds of the way through my second paper. The university which I have enrolled in has a handbook outlining all the criteria for which essays are graded on. In total there are five areas of Assessment Criteria including; relevance, coverage, critical thinking, creative thinking and presentation. Within each of the Assessment Criteria there are five levels of achievement; fail, pass, credit, distinction and high distinction. Each of the levels of achievement has two or three statements of clarification or objectives. All in all the college is VERY explicit in their expectations and success criteria for students.
As I sat down to write my first paper I was understandably a little nervous, it was the first assignment I had written in ten or more years and I was completely naive about the level needed for a masters level paper...although I did have the assessment criteria to comfort me AND the lecturer had taken his time to email me his interpretation of the assessment and exactly what he would be looking for while grading this assignment.
On the return of the paper to my surprise I had done relatively well and did even better than just passing, most definitely due to having a clear understanding of what the lecturer was looking for. HOWEVER, for me the most important part of the reporting back was the good amount of feedback on the areas I which had performed strongly in and the areas I needed to improve on...GREAT! As I began to write my second assignment I had the knowledge of exactly what the lecturer expected of me and the areas which I needed to be more careful in. On the return of the assignment thankfully I found I had improved on my previous grade! Feedforward works!
BUT...then came my second paper. Diligently I sat down to write my second with the comfort an email from my new lecturer to say "Sorry I haven't made contact with you earlier, details about me are in the college handbook, [Lecturer]." Eek, no interpretation of the assessment from his behalf and no idea if the guy had a sense of humour or anything to assist me in writing to him.
Needless to say I was concerned about the grade I would receive as I had little idea as to the lecturer's expectations but was looking forward to the return of my paper so I would gain a little insight to what he was looking for and have a much better understanding of the areas I needed to improve on for the second assignment. Wow, the excitement and anticipation of the return of the paper came crashing down with a generic comment thanking me for the paper however my bibliography was not correctly laid out! FEED ME!!! That was not the worst of it. The worst of it was that the grade he awarded me was not even a grade which the college defined in it's handbook!
I stewed on it for a week or two, not sure what to do and eventually decided that I needed to make contact with the lecturer and asked him for some further information on the areas I did well in (if any), the areas which I needed to improve on for a better grade next time and lastly to highlight the areas of incorrect styling in the bibliography. Unbelievably, the email he returned stated that actually the bibliography was the correct style but that most others completing the paper had not styled the bibliography correctly! What does that mean for me and my baffling grade?
There are times in life which we need to simply learn a lesson and move on! This whole, unpleasant and awkward situation has brought to light the absolute importance of the need for quality feedback/feedforward. Students in our care need direction on how to improve and recognition for areas they are doing well.
Before you sit down and write 'Well Done' in a student's book please consider the scenario above and FEED the STUDENT, it helps them grow!
Has strategy become the new algorithm?
Dianne Scouller M A (Hons); M Ed (Hons).
The Numeracy Development Projects are now well established in our schools. Published
reports record measurable successes, but is all really well in mathematics teaching and
learning in New Zealand schools? Are the successes at the level predicted or of the quality desired? This paper addresses some areas of concern about the balance between the knowledge and strategy strands of NDP, along with other related issues.
There is no doubt that the introduction of the Numeracy Development Projects has had a significant impact on teaching and learning mathematics in New Zealand schools. In the early years of the pilot programme there were promising improvements in achievement and understanding levels for both children and teachers, largely through the involvement of designated facilitators. Has the successful growth of the NDP been its own downfall in a sense?
For many decades mathematics educators have struggled to find ways to help children develop the ability to think mathematically, to make sense of numbers, relationships and symbols, in fact to develop mathematical and statistical literacy. New Zealand curriculum documents have recognised this need, for example Mathematics in the New Zealand curriculum (1992, p.11)was very explicit in this regard stating the goals as being to provide the opportunity for learners to “… develop the ability to think mathematically”. George Booker’s (1999) article Thinking mathematically: problem solving, sense-making, and communicating addressed this question in a way which recognised the complexities involved for both teacher and learner. More recently Neill (2008) addressed the same issues and in doing so raised some other fairly common concerns. He argued for a recognition of the need for the learning, indeed memorisation of basic facts accompanied by plenty practice, rather than a simplistic reliance on strategies. There is no denying the value of strategies, since the memorising of decontextualised facts is barren, to say the least, while at the same time strategies are useless in isolation from an acquired bank of knowledge to which they apply.
Here, in theory at least, the Number Framework has it right, with both knowledge and strategy presented as parallel paths. In the first Numeracy Professional Development Projects booklet it is stated: “It is important that students make progress in both sections of the Framework” (Ministry of Education, 2008, p.1). This paper asks the question as to how effectively that partnership of strategy and knowledge is outworked in the reality of our classrooms, while simultaneously raising some other associated issues around mathematics in our primary schools.
At this point a disclaimer is offered. Much of the concern expressed in this paper has arisen from observations of student teachers and their associate teachers during mathematics lessons, along with conversations with several young teachers managing their own classrooms. Should an in-depth, longitudinal research project be undertaken to investigate the issues raised, the results may be different.
One feature of mathematics teaching and learning in the past was the reliance on an algorithmic approach to arithmetic. The four operations were neatly boxed into procedures guaranteed to lead the student to the right answer if every step was followed accurately. Furthermore, examples to be worked through were often at a computational level far beyond that which was appropriate. Anyone born since the introduction of decimal currency in 1967 and general metric measurements from 1975 can count themselves blessed not to have been confronted with working out problems such as these:
£3-14s-7½ or 51lb-11oz ÷16 or + 5
Nonetheless, for many years the formality of this algorithmic approach predominated, leaving behind those who could not understand what was being done or how and why it ‘worked’. Those who succeeded were largely learners with good memories or those who were able to work out for themselves why they worked.
With the establishment of NDP in New Zealand any attempt to revert to such methods is scorned and loudly pilloried. A very disturbing instance arose early in 2009 where a young teacher was severely disciplined by his mentors for daring to write a problem on the board in the form
The intensity and dismissive nature of the rebuke was alarming.
It can not be denied that exclusive use of these traditional algorithms excluded large numbers of children from success and enjoyment of mathematics, so is not a desirable approach on its own. But was that due exclusively to the reliance on algorithms, or can responsibility also be laid at the feet of traditional methods of pedagogy and the prevailing attitudes to mathematics itself? There is no doubt that some mathematicians firmly reject any idea of an algorithmic approach. Holton (2002, p.23) declared categorically that the philosophical underpinnings of the New Zealand mathematics curriculum are in clear opposition to such an approach:
Right at the start we need to say that we do not see mathematics as a set of algorithms that
have to be learned by heart nor teaching as a process of transmitting knowledge.
His understanding is clearly that the formality of an algorithmic approach prevents children from enjoying mathematics or learning genuine life skills, so the emphasis must be on processes and problem solving.
Let’s look for a moment at this word “algorithm”. The Collins Concise Dictionary defines the word thus: a logical arithmetical or computational procedure which if correctly applied ensures the solution of the problem. If a learner is taught to resolve the problem by saying, e.g. 42 +35 40 + 30 =70 2 + 5 = 7 42 + 35 = 77
what is that but an algorithm? Certainly different from the first illustration, and probably considerably more helpful to the learner, but still an algorithm. Or if a problem is resolved by using doubling and halving, is this not an algorithm too? The answer must be ‘yes’ if the above definition is accurate. It is evident that in some of our classrooms the NDP strategies are being taught in the same rigid way as the rejected methods of the past, rather than being presented as a range of possible procedures designed to enrich understanding of knowledge and help build connections.
Despite the ‘mechanical’ nature of algorithms, their very efficiency is an important aspect some mathematicians believe should not to be overlooked. Some mathematicians believe that when algorithms are coupled with understanding they acquire enormous power releasing learners to draw on cognitive resources necessary for deeper thinking (Akin, 2001; Raimi, 2002; Wu, 1999). There is no contradiction in the NDP Book 1 booklet for teachers, where it is clearly stated
Student should not be exposed to standard algorithms until they use part-whole mental
strategies. Premature exposure to working forms restricts students’ ability and desire to use
mental strategies… However in time, written methods must become part of a student’s
calculation repertoire… (emphasis added)
Basic fact knowledge is crucial. The Number Framework emphasises that the process of deriving
number facts using mental strategies is important in coming to know and apply these facts
(Ministry of Education, 2008, p.14)
There is a development in the learning of mathematics from the fundamental skills to more sophisticated ones which is only possible if those basics have been deeply imbedded in a student’s thinking. Many mathematics educators believe that such an approach is the only way to ensure deep and rich learning of knowledge, relationships and concepts.
Teach the grammar of math first: the facts gained inductively through observations of nature.
Then teach the logic of math: the ordered relationships of facts to each other, the abstract
principles and their applications. Students should be given the opportunity to discuss and
debate mathematical concepts… Finish with the rhetoric of math. Let the students
themselves relate mathematical principles to other areas and to the “real world”.
Nance (1996, p.71, punctuation original)
Yet, deep thinking and problem solving skills must not be neglected by aiming only at the refined development of basic skills (Kulm, 1991). The perceived dichotomy between skill acquisition and understanding of concepts is neatly summarised by Wu (1999, p.1)
This bogus dichotomy would seem to arise from a common misconception of mathematics…
that the demand for precision and fluency in the execution of basic skills in school mathematics
runs counter to the acquisition of conceptual understanding. The truth is that in mathematics,
skills and understanding are completely intertwined. In most cases, the precision and fluency
in the execution of the skills are the requisite vehicles to convey the conceptual understanding.
There is not “conceptual understanding” and “problem-solving skill” on one hand and “basic
skills” on the other. Nor can one acquire the former without the latter.
Let the main things be the main thing
So what are the goals of mathematics education in New Zealand? When the first full mathematics curriculum document Mathematics in the New Zealand curriculum (1992)was published, its main focus was on the problem solving approach. A clear rationale was presented, declaring the value of this approach to be the provision of a way for learners to learn to think mathematically, as mentioned above (MiNZC, 1992). By identifying one of the six strands of the curriculum as Mathematical Processes, with three substrands - problem solving, developing logic and reasoning, and communicating mathematical ideas - the Ministry of Education declared its belief in the need for students to develop the ability to think clearly rather than rely on memory recall alone. Similar goals are inherent in The New Zealand curriculum (2007, p.26).
These two disciplines [mathematics and statistics] are related but different ways of thinking
and solving problems. Both equip students with effective means for investigating, interpreting, explaining, and making sense of the world in which they live.
There is little doubt that the goals of the NDP in outworking these wider goals are valid and valuable in acknowledging that a rich understanding of number is paramount, forming a base on which wider mathematical understandings can be built.
One concern raised by Begg (2006) was that of potential overemphasis on number to the detriment of other branches of mathematical learning, despite a clear indication that “It is important that students can see and make sense of the many connections within and across these strands” (Ministry of Education, 2007, p.26). Has his counsel been heeded? Surely rich knowledge and a broad bank of strategies in number must flow over into other strands to enrich the learning in those areas and so to develop a conceptual understanding of the inherent relationships. Has the goal of a deep and rich mathematical literacy been taken captive by the emphasis on building the foundation?
As mentioned above, theoretically the NDP align both knowledge and strategy, but the challenge this paper poses is that in too many instances strategy overrides the development of appropriate knowledge. This should not be a surprise, as for many years now New Zealand has developed its curriculum on the Outcomes Based Education model which values process above knowledge. There is a certain irony in the declaration of New Zealand being a knowledge economy while our schools are consistently reducing content knowledge in favour of strategies and competencies. It should not be a surprise either that in many classrooms the learning of foundational mathematical knowledge has been taken captive by an undue emphasis on learning strategies.
Neill (2008) does not deny the importance of strategy, nor does this writer, but misunderstanding of its role in the learning of mathematics has given rise to the concerns being expressed. When a teacher lacks the very fluency of understanding he/she wants to impart to the class then it is very easy to use the support material as text books rather than seeing them as the source of ideas and strategies from which a framework for lessons can be formed. This easily leads to a reliance on strategy as the goal of the learning, rather than as a tool with which to develop the understanding of the interrelatedness of mathematical facts and concepts.
It has long been recognised that individual teachers have enormous impact on the quality of learning in their pupils. Those with poor mathematical knowledge are more likely to use a rules-based approach whereas the teacher who has a degree of mathematical fluency is more likely to teach conceptually (Shulman, 1986, cited in Brown & Baird, 1993). More recently Hattie (2005) showed from a meta-analysis of a large number of studies that the teacher’s influence on learning is second only to what the learner brings to the situation himself. This is supported by Ward & Thomas (2008) in their contribution to the 2007 review of NDP, where they recommend targeted professional development for teachers. They raise an interesting issue of teachers’ content knowledge not necessarily correlating with pedagogical content knowledge, suggesting even that a mathematically knowledgeable teacher may still struggle with the balance of knowledge and strategy.
Knowledge versus process (strategy)
Mention has already been made of a perception that the knowledge of basic skills and conceptual understanding are separate from one another. A learner who demonstrates a deep and rich conceptual understanding could be said to have developed a level of mathematical literacy, the goal of the NDP. This leads back to the issue of balancing knowledge and strategy in our programmes to give our students the best possible chance to reach that goal.
In exploring the role of memorisation of basic facts Neill (2008) suggests the need for a learner to have instant recall of these facts in order to work on the problem in hand without the need to redirect attention to access the needed facts. The same can be said of strategies. Each learner needs to have a bank, a repertoire of strategies on which to draw in order to maintain focus on the problem needing solution. So it is clearly not one or the other, but a matter of both truly working together, the knowledge being the material on which the strategy operates.
There is nothing new in this assertion, for many decades ago Skemp outlined advantages and disadvantages of an ‘instrumental’ approach focussing on learning facts and rules, and a ‘relational’ approach focussing on strategies. He believed strongly that depth of conceptual understanding comes only with the recognition of the interdependence of the two approaches. “Knowing how [rules] are inter-related enables one to remember them as part of a connected whole, which is easier” (Skemp, 1976, p.23).
There are some who even believe there is still a place for an element of drill and practice to consolidate the knowledge and skills base in order for learners to develop an automatic response. Akin (2001) contends that any hope of developing real depth of facility or even joy in the art of a subject is vitally dependent on the establishment of a deeply embedded foundation. This is seen to be possible only as a result of constant practice and memorisation, despite a widespread reluctance among parents and educators to include expectations of memorisation from children. Although he acknowledges the challenges associated with deep thinking, he also supports the attitude that algorithms and routines are vital resources which children must be able to access almost automatically.
Neill’s (2008, p.19) concern that “Today there has been a tendency to emphasise strategies” needs something added to it, namely, “at the expense of knowledge and to the point of losing sight of the goal.” No doubt many will challenge such an assertion, but again personal observations of many classrooms in a wide range of schools has provided enough ‘evidence’ for this concern to be voiced. Many a classroom of Year Five and Six classes has been visited where children are only just beginning to learn multiplication tables. Would anyone seriously consider leaving the learning of the alphabet this late?
Raising questions about appropriate pedagogy is another discussion which is not new, but one which has been dominated in recent years by reforms which have rejected the traditional knowledge transmission approach in its totality. Certainly teacher-centred classrooms can be lifeless and dull but they don’t need to be, as Hattie’s (2005) findings clearly show. Current approaches to teaching in general, and mathematics specifically have focussed on child-centred, constructivist pedagogy which has stripped teachers of the opportunity to actually teach, making them merely facilitators. This movement is based on the dubious claim that since knowledge is changing so rapidly we need not actually learn anything, but only know how to access it. Consequently our curricula have developed a major focus on thinking skills to the detriment, or even exclusion of memorising and retaining valuable content knowledge. As Quirk (1998) pointed out, such an approach ignores the fact that remembered content is a vital foundation for all understanding and thinking. This belief in the structure and growth of deep understanding was the basis on which Ausubel developed his subsumption theory of developing schemata (Kearsley, 2004).
Rowe (2007) expressed deep concern at professional ignorance of the impact of effective instruction in creating general education effectiveness, claiming that pre-service trainees are rarely taught how to actually teach, but are instead immersed in constructivist, enquiry-based approaches. His plea is for a balanced approach which recognises the strengths and weaknesses of both approaches. He sees the situation where teachers are not effectively using direct instructional methods for the learning of foundational skills and knowledge as an abdication of professional responsibility. Both Rowe (2007) and Quirk (1998) recognised the implications of this lack for marginalised and underachieving children.
In promoting the wise use of instructional methods Rowe (2007) cites a large body of evidence to support his claim that many cases of underachievement are not the results of cognitive or behavioural challenges, but rather a clear lack of actual teaching and academic challenge. His claim is supported by Quirk (1998) who also believes that children from socio-economically deprived areas have difficulties in school mainly because of the shallowness of content which offers no challenge or inspiration. Recent history has provided several examples to support these claims.
In the 1970s and 1980s Marva Collins worked in the ghettoes of Chicago with African American children, who at five and six years of age had been expelled from school as emotionally and intellectually incapable of learning. Her rigorous academic curriculum led to all her pupils successfully graduating either to high school or into the work force (Collins & Tamarkin, 1982). One major longitudinal study undertaken in the USA is worth discussing, because the results indicate clear success from the application of a curriculum model with a pedagogical focus on direct instructional methods. Project Follow Through was introduced in 1967 with the aim of finding effective ways to counteract the negative effects of poverty on educational outcomes. The project ran until 1995, involving 75,000 children in 180 sites, and 22 sponsors who were given the opportunity to prove that their model of pedagogy and curriculum would in fact successfully raise the learning standards of the children in the study.
The curriculum model mentioned above was that which the University of Oregon sponsored - the Direct Instruction model (DI). Their interpretation of the findings of this study have led them to claim that the data unambiguously proves that their model alone consistently showed significant gains across a wide and variable range of sites. This behaviourally oriented model is a highly controlled and structured program emphasising children’s learning behaviour and mastery of learning (Grossen, 1995/1996; Lindsay, 2004). The advocates of DI also claim that children gain significant cognitive and affective benefits along with the development of academic skills and knowledge. Both Grossen, (1995/1996) and Lindsay, (2004) claim that political embarrassment at the success of this model has lead to it being ignored by educational authorities. Although there are aspects of this particular model which could justifiably be challenged, it can not be denied that the focus on teacher lead instruction led to successful learning for the children.
There are several other groups in the USA currently choosing to reintroduce this instructional approach, including one group of charter schools who have introduced a program called the Knowledge Is Power Program (KIPP). This was set up for much the same reason that Marva Collins began her school, namely deep concern at the poor academic achievement levels of disadvantaged children, again mostly urban African-American and Latino children. Beginning at grade five, these college-preparatory schools set a rigorous program of both extended class time and compulsory homework. Rigorous academic work is balanced with extracurricular programs of music, sport and outdoor field trips. In the 2003/2004 school year large numbers of children from disadvantaged homes, who attended KIPP schools, showed extraordinary progress in reading, language and mathematics (Mathews, 2005). There is a growing body of evidence that the methods used by KIPP schools and others using rigorous, knowledge-rich curriculum models which emphasise teacher directed instructional methods significantly raise the academic achievement levels of minority children, dramatically reducing the proficiency gap between them and those in other state schools (Kersten, 2007).
A caution may be appropriate here, as no programme is ever totally successful, and transferring from one context to another raises its own challenges. However these examples, along with the research of the scholars cited above, offer ample evidence to suggest a very strong case for refocusing our curriculum and reconsidering our approaches to pedagogy, especially in our mathematics classes.
This paper is primarily addressing approaches to the strategy strand of the NDP, but that can not be seen in isolation. Numeracy is only one part of mathematics, and mathematics is only one subject of the entire school curriculum, but the issues addressed with regard to mathematics also apply to all other areas of the curriculum. An option has been presented to consider reinvestigating the value of some aspects of a knowledge-based, teacher-lead curriculum model. In particular this paper poses a challenge to those who teach mathematics at any level to reassess the balance of knowledge and strategy/process. Recognition must be given to the importance of evidence provided by decades of empirical research in support of the need for a rich knowledge base on which learners can build a repertoire of strategies in developing the mathematical and statistical literacy NZC claims as its goal. Along with this is the need to be honest about the extent to which current approaches are failing many of our children. There certainly are successes to be applauded, but far too many children are still failing to acquire the academic and life skills necessary for them to move in to fulfilling career paths.
Akin, E. (2001). In defence of "mindless rote", from http://www.nychold.com/akin-rote01.html
Begg, A. (2006). Does numeracy = mathematics. set. Research Information for Teachers, 2, 21,22.
Booker, G. (1999). Thinking mathematically: problem solving, sense-making, and communicating. set. Research Information for Teachers, 2(8), 1-4.
Brown, C., & Baird, J. (1993). Inside the teacher. knowledge beliefs and attitudes. In P. Wilson (Ed.), Research ideas for the classroom. High school mathematics (pp. 245-259). New York: Macmillan Publishing Company.
Collins Concise Dictionary. 21st Century Edition. (2004). Glasgow: HarperCollins Publishers Limited.
Collins, M. & Tamarkin, C. (1982). Marva Collins’ way. New York: G.P.Putnam & Sons.
Grossen, B. (1995/1996). OVERVIEW: The story behind Project Follow Through, 2005, from http://darkwing.uoregon.edu/~adiep/ft/grossen.htm
Hattie, J. (2005). What is the nature of evidence that makes a difference to learning? Paper presented at the ACER Research Conference, Melbourne.
Holton, D. (2002). www.nzmaths.co.nz. Computers in NZ Schools, March 2002, 23-26.
Kearsley, G. (2004 accessed). Subsumption theory ( D. Ausubel), from http://tip.psychology.org.
Kersten, K. (2007). Teach character to cut racial gap in school results. Star Tribune, Minneapolis-St Paul, 1,2.
Kulm, G. (1991). New directions for mathematics assessment. In G. Kulm (Ed.), Assessing higher order thinking in mathematics (pp. 71-80). Washington: American Association for the Advancement of Science.
Lindsay, J. (2004). What the data really show: Direct instruction really works. Retrieved 02/05/2005, from http://www.jefflindsay.com/EducData.shtml
Matthews, J. (2005a, 11th August, 2005). Study finds big gains for KIPP. Washington Post, p. A14.
Ministry of Education, (1992). Mathematics in the New Zealand curriculum. Wellington: Learning Media.
Ministry of Education. (2007). The New Zealand curriculum. Wellington: Learning Media.
Ministry of Education, (2008). Book 1. The number Framework. Revised edition 2007. Wellington: Ministry of Education.
Nance, J. (1996). Worldview test case. Christianity in the math class. In D. Wilson (Ed.), Repairing the ruins. The classical and Christian challenge to modern education. (pp. 59-74). Moscow, ID: Canon Press.
Neill, A. (2008). Basic facts: start with strategies, move on to memorisation. set. Research Information for Teachers, 3, 19-24.
Quirk, W. (1998). The anti-content mindset. The root cause of the “math wars”. Available: http://www.wgquirk.com/content.html
Raimi, R. (2002). On algorithms of arithmetic, from http://www.nychold.com/raimi-algs0209.html
Rowe, K. (2007). The imperative of evidence-based practices for the teaching and assessment of numeracy. Invited submission to the National Numeracy Review, Camberwell, Vic: ACER
Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Ward, J., & Thomas, G. (2008). Does teacher knowledge make a difference? In findings from the New Zealand Numeracy Development Projects 2007. Wellington: Learning Media.
Wu, H. (1999). Basic skills versus conceptual understanding. American Educator, Fall 1999, 1-7.
Rowe, K. (2007). The imperative of evidence-based practices for the teaching and assessment of numeracy. Invited submission to the National Numeracy Review, Camberwell, Vic: ACER
Ken Rowe’s article is an extremely well balanced approach to issues raised in this paper, as he willingly acknowledges the strengths and weaknesses of both a constructivist approach to teaching maths as well as advocating for a reassessment of the inherent value of teacher directed instruction.
Dianne Scouller is currently the Head of the School of Education at Laidlaw College in Henderson, Auckland, and is currently completing a PhD on a knowledge-based approach to curriculum development. She has been a teacher of mathematics in New Zealand schools since 1968.
Head of School
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It frustrates me as a principal and teacher that at the end of each year/term Christian teachers all over NZ begin the process of planning their coming unit study from scratch! The motivation behind this website was greatly generated out of this frustration.
WHAT IF?? ...Instead of planning from scratch I could choose from 10, 20 or even 100 units which NZ Christian teachers have already designed and taught from!
One of the features of this website is the ability to share documents and resources.
Once you sign into the ChristianEducation.org.nz website a new menu item appears at the very top of the window called [Documents]. This links to a page which gives you the ability to 'upload' your plans or 'download' others plans.
I know every Christian School in NZ is likely to have a different way they plan units therefore it is the content which is important not the layout.
Come on, give it a go! Follow the easy instructions and upload a unit or two.
Let's work smarter!
Justice, Mercy, Humility. Three Pillars of Education.
He has shown you, O man, what is good.
And what does the LORD require of you?
To act justly, and to love mercy
and to walk humbly with your God.
(Micah 6:8; NIV)
Some years ago the concept of considering assessment with respect to justice, mercy and humility was presented to me in a document written by Bev Norsworthy. Since then the idea has brewed somewhat until recently when a discussion with a colleague raised the matter again. That sent me thinking a bit more deeply on the implications behind Micah’s words, and the risk of taking verses out of context straight into a current day application.
The prophet Micah lived in the eighth century BC, a contemporary of Isaiah, Amos, Hosea and Jonah. His prophecies were mainly directed against the leaders of both Jerusalem and Samaria. Those who were at the time leading both Judah and the northern tribes of Israel were charged with corruption and misleading their people. The powerful and wealthy were robbing the poor but there was no redress against such injustices. It was as if the whole of Israel had sold its soul. Micah’s call was for them to return to the commands of their God, because they in fact knew what God required of them. There was no excuse for their lack of obedience and outright rebellion.
What possible connection can such a scene have with education in the twenty-first century? A good starting point would be to consider what Jesus Himself said about such issues, remembering that He said He had come to fulfil the Law. That Law required justice, mercy and humility in all relationships, both public and private. Indeed Jesus spoke harshly to those who were hypocritical in their dealings with others. A succinct summary of His teaching is the “Golden Rule’ of treating others as one would want to be treated.
As Christian educators we do well to keep focussed on the teaching of Jesus. How does His teaching really translate into our classrooms, our policies, our pedagogy? We may well ask whether the policies and practices required of us by educational authorities line up with Jesus’ teaching. Are justice, mercy and humility evident in the national curriculum, in teacher salary structures, in school policies, in classroom management systems, in behaviour policies? Should we even be asking that question?
This verse in Micah has captivated my attention, and the more I ponder on it, the more richly it speaks to me, and the more I want to say, “Yes, we should be asking that question.” It was especially relevant for me in the preparation of a teaching unit on assessment and evaluation for a class of trainee teachers. Very useful discussions have emerged on ways to ensure that assessments are just, that mercy is exercised in appropriate situations, and that both teachers and learners approach their work with a sense of humility and service. It is not difficult to extrapolate the concepts to education in general, and even to life itself.
It is worth remembering, that the word ‘education’ comes from the Latin verb ‘educere’ meaning ‘to lead out’. God lead His people out of slavery in Egypt into Canaan, and through the prophets He was calling His people again, out of the darkness of corruption and evil, and into the light of true obedience and worship of the one true God. He still calls us to come out of darkness and into the light of full salvation in Christ. Relationship with the Living God is outworked daily in all we do, and this includes the teaching and learning we call ‘education’.
To explore the subject in any depth may well lead to a whole book, so this brief sketch will do no more than propose thoughts and hopefully raise questions for further investigation and study. There is no attempt to find verses to justify what is said, but rather general biblical principles are presented in the hope that readers will search the scriptures for themselves.
Justice. What does it mean? Is it the paying of penalties or reparations for crimes committed? Maybe society’s legal system shows us this face of justice more often that any other. But what is the biblical concept of justice? It is clearly something God expects us to do. There are Old Testament passages which speak of fair weights and honest scales, of treating people with respect and dignity, of setting reasonable boundaries, both literal and metaphorical, and holding people accountable to these boundaries. Furthermore we are to realise that God Himself models those expectations for us. What we often forget in this regard is the universal law of consequences. Deuteronomy chapter 28 is the best illustration of this law of reaping and sowing.
With respect to education we could ask whether it is just to withhold from children the body of knowledge common to literate society. Some great Christian scholars have declared their belief that excellent education is to be presented in an integrated framework of truth, deeply grounded in Creator God and the creation. Such an education allows for the development of a rich understanding of the associations between all disciplines and roots all learning in an eternal framework of truth.
Whether education takes place in a school setting or at home, these aspects of God’s justice must be in evidence for rich, holistic learning to take place. Teaching methods, learning activities, assessment tasks, behavioural expectations and relationships all need to demonstrate justice as God defines it. Fairness, respect, setting high standards, following through on consequences, both positive and negative, is vital to ensure that learning is enhanced and relationships nurtured.
The outworking of the law of consequences inevitably involves mercy. Without mercy, justice easily becomes legalism. It must be remembered that the teaching & learning relationship is just that – a relationship between the learner and the teacher, and between each of them and God. It’s a kind of three-strand cord. With God’s grace we can find ways to hold people accountable and still exercise mercy. Without consistency and the demand for accountability we can actually do harm. Some might say we sin against the learner and maybe cause them to sin. Jesus had very strong words to say to those who cause children to sin.
This, then, is where humility plays its part. The apostle James cautions us to realise the enormous responsibility and wonderful privilege it is to be a teacher. That might mean in a school. It might mean at home or at church, or in some other environment. But we do well to remember that much (accountability) is required of those to whom much (responsibility) is given. So we approach the task of teaching with an understanding of the genuinely awesome joy and privilege it offers. We will never be truly fruitful teachers unless we learn to humbly rely on the LORD for His strength, His wisdom, and His grace. This does not preclude our own responsibility to learn the necessary knowledge and skills and to develop the appropriate attitudes to fulfil our calling. But it does mean that we don’t have to try to carry out this role of teaching on our own. Humbly submitting to the sovereignty of God will bring the grace to submit to authorities, to treat colleagues and families and learners with respect and to acknowledge that we are but instruments in the Master’s hands. Very valuable and important instruments, but instruments, none the less.
Justice, mercy and humility – three pillars of education, built on the foundation of Christ and His Word.