NUMERACY LEARNING AND TEACHING
PART A: Initial Assessment and Goal-setting
Assessment is essential in any form of learning. In numeracy learning and teaching, assessments can either be initial or diagnostic. Initial assessment refers to the process by which a teacher seeks to find out about the learner as an individual and to know the skills and abilities that the learner possesses (Wright et al., 2006 pp.33). Gal and Ginsburg (1994 pp. 4) argue that initial assessment helps a teacher in determining the expectations and motivations of a learner as well as well as determining their capability to achieve in a certain subject area. Conversely, diagnostic assessment evaluates a learner’s knowledge, strengths, skills, and areas that require improvement with a focus on a particular subject area (Higgins and Parsons, 2009 pp. 236). The purposes of diagnostic assessment as identified by Higgins and Parsons (2009 pp. 236) are: to ascertain the learning preferences of learners, ensure learners access the required support, identify previous experience, and identify the knowledge gaps that need to be bridged in the teaching.
Initial and diagnostic assessments are carried out by asking specific and objective questions (Treagust, 1995 pp.348). Some of these questions have been structured and standardized into specific tests for initial or diagnostic assessments. Usually, the teacher is the person mandated to carry out the tests in accordance with the regulations of the learning institution (Treagust, 1995 pp.348). The tests can be done before learning commences (Treagust, 1995 pp.348) – during the reporting week or online before the learner joins the institution. Online assessments are advantageous in that the learner takes them in the comfort of their home and is not frightened by too many tests on the first week of reporting to an institution. Quality and access to these tests can be enhanced by the creation of better online platforms. Such platforms will enable the learners to interact with the facilitator conducting the test and to clarify contentious questions or phrases while taking the test. This will also enhance the quality of results that the test gives as the result will be a true reflection of the learner’s abilities and not a product of confusion.
Diagnostic assessment is a retrospective assessment that is meant to ascertain the knowledge and skills that the learner already possesses (Patel and Little, 2006 pp. 133). Diagnostic assessment can also be used to understand the nature of difficulties that a learner is facing and to determine the best way of helping the learner overcome them (Patel and Little, 2006 pp. 133). The main aims of diagnostic assessment inform its timing; it is done either before learning commences or after difficulties arise. Since diagnostic assessments are mainly done before learning starts and before the learner and the teacher familiarize, the language used is formal and simple. The questions in the tests should be straightforward, short, and precise.
As earlier alluded to, the policy of an institution determines the type the diagnostic assessment test given (Nichols, 1994 pp. 579). First are the predesigned commercially available tests – these are highly objective and are standardized. Second, institutions can design their own diagnostic assessment tests and utilize them. Finally, some institutions would prefer facilitators organizing interviews with learners – in the interviews, the facilitator has the liberty of formulating questions there and then (Nichols, 1994 pp. 580). Such interviews have the advantage of retrieving as much information as possible but a lack of standardization is a major pitfall.
In numeracy learning, diagnostic assessments look for preexisting knowledge and arithmetic skills. For knowledge, learners can be asked of their experience in topics which they should have covered at their level (Leighton and Gierl, 2007 pp.14). For skills, the learner should perform tasks like drawing geometrical shapes to ascertain whether they have skills that they should have acquired at their level.
Standardized and detailed diagnostic assessment tests give the most valid results for numeracy learning (Nichols, 1994 pp. 581). However, this may not determine the nature of difficulties for learners who are assessed after facing difficulties. In that case, interviews, which give the facilitator liberty to formulate questions they deem appropriate there and then are the best. The results of a diagnostic test can inform the teacher how to handle the learner and whatever difficulties they are facing. These results make more meaning and are most beneficial to the learner when the teacher shares them with the learner exclusively (Tunstall, and Gipps, 1996 pp. 196).
Online diagnostic assessments are easy to administer and mark. The only cost that might be incurred by an institution is that of buying the predesigned tests. Its, therefore an efficient test. One advantage of using the predesigned diagnostic assessment tests in their proven content and construct validity. Moreover, the reliability of such tests can be improved by making the questions closed ended – this makes the marking of the tests standardized.
After initial and diagnostic assessment is done, goals are set in order to address the various issues that arise. The specific purposes and processes of goal setting are discussed below. First, goal setting helps the teacher plan. In numeracy teaching, the teacher has to plan when and how to deliver the topics based on the results of the initial and diagnostic assessments (Wright et al., 2006 pp.153). Second, it helps to make learners focused on a particular area that needs to be covered within a given period of time. Thirdly, setting goals based on assessment results is meant to improve the weaknesses that have been identified in the learners and to motivate the learners (Meader, 2000 pp. 9). Finally, well-set goals that critically consider the assessment results help a teacher to complete a topic on time and to complete the topic effectively. Moreover, since the assessment results can help in grouping learners based on knowledge level and ability, properly set goals will ensure that topics are delivered effectively to the various groups of learners.
The process of goal setting starts with breaking the topic to be delivered into smaller steps (Ness and Bouch, 2007 pp.7). This breakdown is mainly dependent on the level of preexisting knowledge in the learners as demonstrated in assessment results. The core curriculum is key in this breakdown and throughout the whole process of goal-setting. Whatever methods and strategies are chosen should be in line with curriculum demands. Next, the teacher should map a journey that will lead to mastery of the topic. This will rely on building on what the teacher knows the learners already know. Finally, the goal-setting should outline exactly how the teacher shall assess themselves at the end of the topic.
PART B: Principles of Effective Numeracy Learning and Teaching
Statistics are important in numeracy teaching at all levels. Numeracy has been defined as the effective use of mathematics to meet general demands of life. Teaching statistics in the right way can help learners to find this seemingly difficult topic much easier.
Six principles underpin effective numeracy learning and teaching. The first principle deals with the articulation of goals (Sullivan, 2011 pp.25). As earlier alluded to, the results of initial and diagnostic assessments are key in the process of goal-setting. In teaching fractions, it is important for the teacher to interact with the learners closely, to tell them the tasks that they will be required to perform, and to let them know how he or she expects them to learn.
The second principle is making connections (Sullivan, 2011 pp.25). A teacher can make connections with what the learners already know hence emphasizing the importance of diagnostic assessment. Connections are also meant to show the rationale of teaching a certain topic and enhancing mastery of content at the end of the topic. In teaching statistics, the teacher should look to create connections between real life situations and classwork (McLeod and Newmarch, 2006 pp.12). Moreover, numeracy teachers should look to link the topic of the day to other topics in the curriculum so as to ensure a continuum of knowledge.
The third principle that is important in literary teaching is fostering engagements (Sullivan, 2011 pp.26). For learners to master the principles being taught and to utilize them properly in real life, they should be engaged in the teaching process. The main tasks that can help learners to master fractions are drawing and computing. Learners can be asked to draw graphs and charts using a particular set of data so as to make more sense with the data – to turn data into information. Computer technology can also be utilized to draw the graphs and charts. Computer technology confers an extra advantage of making the figures three-dimensional hence enhancing understanding (Lugalia et al., 2013 pp. 4100). This will require the teacher to avail resources like manila papers on which the learners can draw. An activity like drawing can relieve learners of boredom that is associated with continuous classwork. The teacher should also choose problems that the learners can engage in computing. Here, the teacher should emphasize more on the method of arriving at an answer rather than the answer itself.
In addition, incorporating real life situations like division of resources or computation of class marks can help learners to understand statistics much better (Furner et al., 2005 p.19). Here, computer technology can be very useful in making virtual real life situations (Lugalia et al., 2013 pp. 4100). There are computer programs, which can allow for virtual manipulations, for instance, the division of pizza. As Hollingsworth et al. (2003 pp.21) say, activities will allow learners an opportunity to show the teacher their own thinking; with these the teacher can know how best to continue with the topic. It is important for the teacher to get the views of learners in the teaching process as the current curriculum emphasizes learner-directed learning and teaching. Leach and Moon (2000 pp.389) argue that integration of computer technology into the learning process benefits both the teacher and the learner massively.
The fourth principle of effective numeracy teaching is differentiating challenges (Sullivan, 2011 pp.27). For effective numeracy teaching, a teacher should determine the challenges that are facing the learners. Diagnostic assessment is a key tool in this principle. Apart from diagnostic assessment sessions, teachers should also look to develop the closeness to the learners in order to identify the problems facing the learners. Moreover, adoption of a learner-directed learning system can aid in the identification of challenges. It is also key for the teacher to encourage collaboration between learners (Marat, 2007 pp.209). In terms of teaching statistic, learner collaboration is a key pedagogical approach. The teacher can do this by dividing the learners into groups and assigning questions to the groups. In this case, mixed ability groups will yield the best results, as the group will put together learners with various abilities.
Collaborative teaching also involves close collaboration of teachers in a particular department. By sharing experiences and knowledge in delivery of a topic like statistics, performance of the whole department is improved. Encouraging collaboration among students benefits the teacher and the entire learning process massively. Instead of dealing with individual students, the teacher has the luxury of dealing with numerically fewer groups. Group-work also enhances better mastery of a topic as the sharp learners in a group can explain concepts to those of lesser ability in the same group (Swan, 2006 pp. 235). Groups also provide a platform for shy learners and learners of low ability to ask questions and to showcase whatever they can do; Patel and Little (2006 pp. 134) conclude that peer study groups can avert anxiety in individual children. One key principle for better delivery of content on statistics is encouraging learners to question; group work can encourage learners to question, and to do it in various ways. The best groups are selected after the teacher has interacted with the learners and ran initial and diagnostic tests. According to Swan (2006 pp. 238), working in groups enhances performance and motivation in learners.
Fifth, properly structured lessons are key to mastery of content in numeracy teaching (Sullivan, 2011 pp.28). Sullivan (2011 pp.28) asserts that a proper lesson has four parts: It starts with an introduction which lays a firm basis for the rest of the lesson and reveals the rationale of learning this particular lesson for the learners. The teacher then delves deep into the topic of the day – for statistics, this will involve giving problems to be tackled by individual learners and in groups (Sullivan, 2011 pp.28). Then an explicit class discussion follows – in teaching statistics, the individual students or groups who tackled the question posed present their working before the whole class and a discussion follows, questions from students can be answered at this point; the teacher then wraps up the lesson with a summary of the key ideas. Again, the results of initial and diagnostic assessments which were carried out earlier in the semester are essential (Sullivan, 2011 pp.28). These results inform the approach of the teacher, especially what the teacher needs to give as an introduction to the topic.
While the learners are tackling the posed questions, the teacher needs to be there to assess their progress. In such assessments, the teacher might discover the mistakes in the learners’ work or the areas that they failed to understand. Learners find teachers who offer them such assessments to be better (Swain et al., 2005 pp.84). The teacher can then use this as an opportunity to lay more emphasis on the areas that seem not to have been understood well in their summary.
The last principle of ideal numeracy teaching is concerned with promoting fluency and transfer of knowledge (Sullivan, 2011 pp.29). The teacher can ensure this by assessing the learners regularly. Through assessment, the teacher can learn more about the difficulties that the learners face. One way in which a teacher can ensure regular assessment especially in numeracy teaching is by asking questions at the end of a teaching session or by giving assignments at the end. By asking questions at the end of a session, the teacher assesses not only the learners’ grasp of the topic but also his or her own delivery of the content. It is important for a teacher to also seek feedback from learners. The best and honest feedback from learners can be retrieved by giving anonymously filled forms to learners at the end of a particular topic.
PART C: Promoting Numeracy and Wider Skills in Learning Programs
Curriculums in most areas of the world look to integrate the major areas of learning – language, mathematics, and science. This integration can make learning easier, more enjoyable, and easily applicable to real-life situations. Incorporating numeracy into other areas of the curriculum enriches these areas (Sullivan, 2011 pp.21).
In teaching English and literature, statements with mathematical meaning could come up. Explaining these mathematical sentiments in the best way come uncover their real relevance. Moreover, in uncovering the real relevance of such statements in literature, teachers can use that as an opportunity to teach some concepts of numeracy (Atkin et al., 2005 pp. 95). Further, aspects of teaching English and other languages can be used to help learners to understand numeracy better. Approaches like reading texts together, identification of vocabulary and writing them down, and suggestion of synonyms can really help learners in both literary and literacy learning. Such approaches have proven to be successful for ESOLs who are also learning numeracy (Atkin et al., 2005 pp. 95).
This, therefore, means that the integration of numeracy into language can be a form of embedded numeracy provision. Good teachers always look to integrate numeracy into literacy classes like language classes – this makes the classes more interesting and makes it easy for the teacher to drive several points home at the same time effectively (Swain et al., 2005 pp.83). The effectiveness of embedded numeracy provision as in this case is highly dependent on the credibility of the results of initial and diagnostic assessments that had been conducted earlier. Moreover, a better needs analysis needs to be conducted so as to determine just how much the embedding needs to be done. Targeting is also key here; teaching interventions need to be clearly focused to prevent the teacher from veering off track in their delivery.
When done well, embedded numeracy provision can be very effective. It enhances learner participation in the learning process. As earlier alluded to, by embedding numeracy teaching in other subject areas in the curriculum, a teacher can cover more topics in a much shorter time (Casey et al., 2006 pp.37). By combining language and numeracy teaching, the teacher can easily use the words of the language to belabor concepts in numeracy or use concepts in numeracy to bring to the attention of learners the real meanings of various phrases in literature (Atkin et al., 2005 pp.72). In addition, embedded provision evades the boredom that is experienced by learners after being constantly bombarded with various concepts of a single subject area; a few concepts from a different area are thrown in during the class to recapture the attention of learners (Roberts et al., 2004 pp. 90). The only downside to embedded numeracy provision is the fact that the key concepts being brought out in the lessons might be too many for the learners. Embedding numeracy with language will mean that in one class, the learner has to grasp a number of language concepts and a few numeracy concepts as well; with time, this might prove to be too much and the learners will grasp little.
Embedded numeracy provision is particularly good for ESOLs. This strategy helps them to master content in both numeracy and literacy much faster. Embedding numeracy and other areas of curriculum for ESOLs can be done by partnership in teaching and by vocational teaching.
Discrete numeracy provision is different in that, here, the teaching is focused on a particular area of numeracy. The teaching is usually structured in short courses dwelling on a particular numeracy skill that the learner wishes to acquire (Office for Standards in Education (England)(Ofsted), 2011 pp.30). Discrete numeracy provision is mainly meant to enhance numeracy skills, especially in adult learners as these skills have proven to be very useful in recent times.
Discrete numeracy provision usually focuses on small subject areas that are key in a community, for example, cooking on a budget or money matters. It is the main approach in adult numeracy provision which is meant to enhance mathematical skills that are required for various careers (Griffiths and Stone, 2013 pp. 31). It is possible to integrate other areas of learning in this form of numeracy provision but only to a limited extend. The other main challenge of discrete numeracy provision is its failure to consider the results of initial and diagnostic assessments. The courses are usually predesigned and are rigid this cannot be modified easily by the teacher so as to suit the needs of a particular learner.
Discrete numeracy provision offers one advantage, though; since the teaching is highly focused, the number of concepts to be taken in by the learner is reduced hence allowing time for the learner to take in the available concepts quickly. However, the method has not been as effective as results achieved in the past have not been impressive (Office for Standards in Education (England)(Ofsted), 2011 pp.31). Discrete provision is particularly tough for ESOLs since the numeracy content is mainly delivered in English, lack of simultaneous English lessons can make learning difficult. Embedding remains the best way of numeracy provision. Most learners also choose embedded over discrete numeracy provision (Atkin et al., 2005 pp.72).
One advantage that embedded numeracy provision has over discrete numeracy provision is the ability of the latter to use language to help students in understanding and remembering vocabulary in mathematics (Griffiths and Stone, 2013 pp. 78). Vocabulary in some areas of teaching fractions can be challenging (McLeod and Newmarch, 2006 pp.11). For instance, it might be hard for learners to differentiate between ratios, fractions, and proportions. The distinguishing between numerators and denominators can also prove to be quite a task for many learners. Vocabulary is, therefore, a critical part of teaching fractions.
Apart from embedding numeracy teaching in other subject areas, a teacher can use other strategies to enhance mastery and memory of various vocabulary in teaching fractions. The use of word-walls is a one of these – a word-wall refers to a particular section on the wall of a classroom where all vocabulary are written. The learners are reminded of the vocabulary anytime their eyes look at the word wall – this keeps the vocabulary in the minds of the learners thus enhancing mastery and memory. Another strategy is the use of graphic organizers. The Frayer model graphic organizers are by far the most common of these. These usually utilize a triangular shape divided into four portions with space for the vocabulary at the center. Each of the four portions contains various aspects of the concept like definition, examples, and other facts. These two are visual aids which can help in mastery of vocabulary.
For ESOLs, embedding numeracy with literacy is the best way to aid in mastery of vocabulary. Literacy strategies like spelling and use of key phrases can be employed in this case (Furner et al., 2005 p.19). It is crucial for teachers to also encourage learners to use the vocabulary they learn regularly that is learners should learn to talk mathematics.
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