Literature Review Outline
ECD 503: Introduction to Research and Evaluation in Education
I. INTRODUCTION 3
II. MATHEMATICAL JOURNALING (FIRST INTERVENTION) 4
WHAT IT IS—DEFINITIONS AND DESCRIPTIONS 4
EXEMPLARS OF MATHEMATICAL JOURNALING WITH POSITIVE OUTCOMES 6
ARTICLE 1 6
ARTICLE 2 9
ARTICLE 3 11
SUMMARY OF MATHEMATICAL JOURNALING RESEARCH 13
III. CLASSROOM DISCUSSION TO DEVELOP MATHEMATICAL THINKING 15
WHAT IT IS 15
EXEMPLARS OF DEVELOPING MATHEMATICAL THINKING 15
ARTICLE 1 15
ARTICLE 2 17
ARTICLE 3 19
SUMMARY OF DEVELOPING MATHEMATICAL THINKING THROUGH DISCUSSION 21
IV. QUESTIONING AND SCAFFOLDING STUDENTS’ THINKING 22
WHAT IT IS—DEFINITIONS AND DESCRIPTIONS 22
EXEMPLARS OF SCAFFOLDING STUDENT THINKING WITH POSITIVE OUTCOMES 22
ARTICLE 1 22
ARTICLE 2 24
ARTICLE 3 26
SUMMARY OF SCAFFOLDING STUDENT THINKING RESEARCH 27
For a long time, mathematics as a subject has been given moreattention by both teachers and students compared to other subjects.Two surveys by Gallup poll, one conducted in 2002 and the other in2013 in the US showed that mathematics was considered the mostvaluable subject way ahead of English, science and history (Jones,2013). A total of 34% of the respondents found the subject to be themost valuable in both surveys. However, the perceived value ofEnglish/literature/reading declined from 24% in 2002 to 21% in 2013(ibid). If these findings were to be followed, teachers should accordmore importance to math than other subject. Consequently, thestrategies in teaching mathematics are likely to be more scrutinizedthan other teaching strategies in other subjects.
A number of strategies in teaching mathematics have been developedfor various levels over the years. These strategies are informed bycountless years of research and empirical observation. Somestrategies have been level specific. For instance, Ginsberg et al.(2004) note that there is little attention given to math teachingstrategies for preschoolers. The author thus suggests a course fortraining teachers in which the psychology of mathematics isemphasized in the very first weeks for preschooler learning. The ideais to increase interest and highlight relevance of mathematicalongside other subjects. For some other teaching strategies, theyare suited for specific tops in mathematics such as geometry oralgebra. In the context of this paper however, interest is ongenerally applicable mathematics teaching strategies on a wide rangeof topics. Three of the main strategies are mathematical journaling,talking about mathematics and questioning.
B. Implications based on research mentioned
The benefits accrued from competent mathematics teaching strategiesnot only deliver on demand education but also improve and enhanceperformance in mathematics. It is estimated that good performance inmathematics, going by the importance attached to it is likely toboost confidence academically. Therefore, it is very important forstudents to perform well in mathematics. This can only be realized ifteachers apply working methods and strategies. This has far reachingimplications on the formulation of curriculum for student teachers invarious levels and especially for the at risk children. Thesechildren need special attention and a probably a more intensiveapplication of the mentioned strategies.
II. MATHEMATICAL JOURNALING (FIRST INTERVENTION)What it is—definitions and descriptions
Mathematical journaling is a process whereby students write down thelearning process triggered by several prompts such as a before atask, reflecting on a completed task or an ongoing one (Liljedahl etal. 2007). This can involve writing down procedures and the thinkingprocess that explains an action or process in mathematics. In somecases, journaling is also referred to as writing learning logs(Wilcox & Monroe, 2011). Mathematical journaling is therefore,the writing down of the thinking process rather than method thatexplains given steps and procedures and also challenges encounteredin accomplishing a task. In mathematics, therefore, it involveswriting down procedures and the thinking process used in solving aproblem. Liljedahl (2007) however, notes that the process of problemsolving in mathematics is largely unclear and involves many dead endswhich the leaner might be unwilling to document and hence there aredifference in the actual problem solving process and what ispresented and documented in the journals. The National Council ofTeachers of Mathematics (NCTM) has instituted journaling or writingin mathematics as a key requirement in teaching mathematics (Kostos &Shin, 2010). This in informed by some of the noted benefits ofjournaling as discussed in the next phase.
Thereare generally two types of journals those that can be revised andaccessed by other parties and private ones that are kept by studentsalone for private use. For the private journals, the journals are notrevised by anyone else apart from the students. For the revisedjournals, they can be assessed by teacher or other students. Thoseexchanged amongst students, individual students are expected tocomment on their colleagues work. This of commentary engages studentsin healthy debates where one takes the role of teacher by seeking tocorrect or concepts to fellow students.
Benefits of mathematical journaling
Severalteachers, researchers and scholars have sought to apply mathematicaljournaling and identify its benefits. One of the key benefits ofjournaling as a cross-curricular benefit is improvement incommunication capabilities. Students learn to explain themselvesbetter and are also able to improve on their writing abilities byimproving technical vocabulary in mathematics (Liljedahl, 2007).
Mathematicaljournaling fosters understanding of mathematics. This is observablein the sense that learners seek to better explain and be understoodby detailing the thinking processes involved rather than memorizingof the method used in solving the problem (Kostos & Shin 2010).Wilcox and Monroe in their various studies noted that one teacherobserved that in one class, one of the students defined basicprobability and went as far as providing clear and unique examples.This shows that journaling or keeping mathematics logs engage theminds of students and assist them in transforming of information frombeing memorized facts to construction of meaning (ibid).
Exemplars of mathematical journaling with positive outcomesArticle 1
Objectives and hypotheses
One ofthese studies was carried out by Huang and Normandia (2007) in asecondary school in New Jersey. The study adopted a functionalapproach to assess the linguistic methods used by students to gainmathematical understanding both in solving questions andunderstanding mathematical concepts. Specifically, the researchsought to find out the typical linguistic methods used by students tounderstand mathematical knowledge pertaining to procedures andconcepts. The researchers hypothesized that language competenceinfluences mathematical competence.
The study relied on a stratified sample of 25 students from one ofthe private high schools in the central region of New Jersey. Thestudents were taking a standard pre-calculus level class. All therespondents were native English speakers from a mixed backgroundcomprising of wealthy and middle income earners. The respondents,studying in two different blocks, had a history of average academicperformance. The two different groups of students shared one teacherwho had more than three years of experience in teaching.
As percurriculum requirements, all students were subject to learningwriting across the curriculum. In the classroom, students wereregularly required to demonstrate and explain to fellow studentsvarious mathematics procedures orally. A total of 5 lessons eachlasting 77 minutes were observed on-site and notes taken. Theteaching methodology and the context of the writing assignment werenoted. Additional formal voluntary interviews were conducted with asample of 22 respondents from the two blocks. The interviews centeredon the students’ perception of the relationship between the writingtasks assigned and learning mathematics.
Theintervention in this case was a lesson and assignment on thequadratic formula. The assignment required the students to explain inprose the correct procedure of using the complete square technique tosolve a standard form quadratic equation. This prose form wasrequired to have the three maim essay sections introduction, bodyand conclusion. The algebraic process was also to be displayed in adifferent paper. The assignment was to be completed in two and halfweeks. A sample of 11 assignments was picked with consent fromstudents for analysis after the teacher had marked the papers.
All thestudents scored between 45 and 49 out of a possible 50. Deanna hadthe lowest score of 45 while Jenna scored the highest, 49. The maindifference between the two was that Jenna laid emphasis on theprinciples used in the process and the steps while Deanna laidemphasis on the sequence of steps. On knowledge structures, Jenna wasclearer on it structuring. She first defined an equation, and noteddifferent forms of equations among them quadratic equation. Shefurther noted that there are different types of quadratic equations.She explained there are various ways to solve quadratic equationincluding the quadratic formula. She followed the procedures. As forDeanna, she only highlighted quadratic equations as one of the manytypes of equations. She went straight to methods of solving it. Sheindicated the completing the square as the method used. Jenna furthershowed the effect of each step and how each linked to the next whileDeanna did not.
Onlinguistics, Jenna and Deanna followed different paths. For instance,Jenna made 23 sematic relations to Deanna’s 14 while both used asimilar number of different linguistic devices at 16 each. However,Jenna utilized more linguistic features at 65 compared to Deanna’s51. Jenna was more capable of expressing cause-effect relations atseven compared to Deanna’s only three.
Thewriting processes and procedures taken by the different studentsinfluence the score hence performance. It is also clear that betterwriting skills in the procedures involved fostered betterunderstanding as indicated by the ability to link each step to thenext in the case of Jenna.
Thestudy relied on a small sample that enabled it to be more detailedand collect more informative data. On the other hand, the study didnot explain other aspects that may have altered the score obtained bythe students such as grammatical errors and syntax problems. Thestudy also failed to highlight how the observations were made andwhether the teacher was involved in making on-site observations inclass.
Objectives or hypotheses
Asecond study by Kostos and Shin (2010) took an action researchapproach to find out how mathematical journaling affected secondgrades’ students communication in mathematical thinking. The studyhypothesized that mathematical journaling as a teaching strategyimproved mathematical thinking capabilities and thus performance inmathematics and even other subjects that require writing.
As anaction research, the study relied on a given classroom of secondgraders to act at the subject of study with the teacher as theresearcher. The class comprised of students of mixed abilities inlarge suburb school in Chicago comprising a population of 640students starting from kindergarten to fifth grade. The target classcomprised of 20 students but four were disqualified for the study.Eight boys and eight girls were involved. Ethnically, theparticipants were two Asians, one African America, two Hispanics andeleven Caucasian. The researcher/teacher has over 12 years teachingexperience with two as a grade two teacher and five a chairperson ofthe district mathematic curriculum.
Theinstruments used were direct on-site observation. The teacher was theresearcher and thus made all observations and took down notes. Datawas also gathered from “(1) pre- and post- math assessments (2)students’ math journals, (3) interviews with students and (4) theteacher-researcher’s reflective journal” (Kostos and Shin p.226).To preserve participant’s identity, pseudonyms were issued to allparticipants.
Theintervention used in the study involved issuing of identical mathassessments from the Illinois State Board of Education at thebeginning of the study. The teacher taught the instructions for aperiod of five weeks in which students were required to make theirjournal entries. A total of 16 prompts were used resulting to anaverage of three journal entries per week. The prompts were conceptsand topics previously covered in class which included grouping,addition and subtraction. The teacher modeled for the students forthe first three prompts in the first week. The modeling showeddifferent strategies such as drawing, making charts and writingnumber sentences. In the remainder of the four weeks of the course,students made 13 math journal entries. Three mini lessons were alsotaught during this period which largely assisted the students inunderstanding the math problem.
Theresults from the test were analyzed together with the mathematicaljournal entries made by the students. Other results were collectedthrough interviews from eight of the sixteen students. Fourthly, theteacher wrote a reflective journal on the mini lessons.
Results from both pre and post assessments scored recorded anoteworthy statistical difference. In the pre-assessment, mean scorewas 7.25 out of 12 which increased to 10.0 in the post assessment.Explanation scores in the pre-assessment journals averaged 1.13compared to 2.56 in post assessment score. Mean scores in the journalentries rose from 0.88% to 1.19% in the second attempt. For groupedjournals, mean score increased from 1.00 to 1.33. The teacher’sreflective journal which reflected on the mini-lessons recordedimproved changes in students’ attempts to explain themselves. Thestudent math journals, student interviews and the teachers reflectivejournals all recorded significant improvement in mathematicalvocabulary.
Journalsassist in the learning and practicing of mathematical vocabulary.Other than that, when journals are used a communication tool withteachers, they help highlight the problematic areas and concepts thatthe teacher might want to teach again and explain further if a commonproblem is observed. Alternatively, they can be helpful in assistingpersonalized training as teachers can identify the specific areas ofweakness facing individual students.
Thestudy is very positive in that it addresses a sample group that isoften ignored when it comes to mathematical journaling. Secondgraders are likely to be perceived to be too young to engage inmathematical journaling given that their writing skills are stillunderdeveloped.
Objectives or hypotheses
Thethird article that highlights mathematical journaling is by Wilcoxand Monroe (2011). This study used secondary data to assess thetechniques used in journaling that can be incorporated in mathematicsand their impacts on individual students. The study, as informed bysimilar previous studies, hypothesized that writing/journaling inmathematics improved performance both in mathematics and writtensubjects.
Thestudy sampled various findings from several studies. The samples usedin the various studies ranged from 3rd, 5th and4th graders in various schools. The researcher does notprovide the actual sample size. However, it is indicated that all theresearchers in the several studies assessed were respective teachersof the various classes.
Theteacher observed performance of students on assigned tasks in amathematics lesson. The amount answers provide by the students andtheir responses in the mathematics journaling attempts were assessedfor any recognizable trends on thought processes leading to thesolving of the problems and the writing skills.
Theseobservations were made on a wide range of assignments given in thedifferent classes. The topics covered included algebra, geometry,fractions, probability and statistics. The prompts were in form ofassignments and simple tasks given in class. All the teachers hadpreviously and extensively covered the topics in class prior to theprompts.
Theresults from the various studies indicate that teachers employvarious techniques in journaling. The strategies were loosely groupedinto two writing with revision and writing without revision. Thechoice of strategy chosen appears to vary with the grade. Forinstance, 3rd graders were allowed to use images toportray numbers as they pictured them in their minds. The studyrevealed that writing without revision such as making notes fromnotes taken in class, writing learning logs and writing down answersrather than answering orally questions asked in class all improvedunderstanding of mathematics, use mathematics vocabulary andlanguage/writing skills. Writing with revision included sharedwriting, class book and alphabets books. It also had similar benefitsin addition to enabling teachers to direct future lessons better andto identify problematic areas that need to be revised.
It isclear that there is no universal method of journaling that isprescribed to specific group of leaners. However, from the strategiesof mathematical journaling assessed, teachers should choose the bestapproach according to their leaners needs.
Thestudy offers several images from the actual students responses fromthe various studies reviewed. In doing so, it clearly illustrates theexpected results in employing the various journaling strategiesreviewed. However, the study does not provide a clear explanation onthe methodology used in settling for the various studies. It isexpected some experiments with mathematical journaling encounteredproblems but it is not indicated.
Summary Of Mathematical Journaling Research
Overall summary and synthesis of benefits
Mathematicaljournaling improves cross curricular writing skills. Given thatmathematics is largely associated with calculations, little attentionis given to the language requirements of mathematics. Mathematicaljournaling improves language skills in mathematics and enhancesunderstanding of mathematics concepts. This is because journalingexercises self-expression and reflective skills which are applicablein other disciplines.
and synthesis of instruments
To bestunderstand the effectiveness of mathematical journaling, actionresearch approach should be used together with a mixed methodology.Kostos and Shin cite Mills (2003) who define action research as ‘‘anysystematic inquiry conducted by teacher researchers… to gatherinformation about… how they teach and how well their studentslearn’’ (p.5).
and synthesis of interventions
Classworkassignments, homework and tests are the main key interventions inteaching mathematical journaling. From the study, it is clear thatthe efficiency of any study and for the credibility of the findingsand the whole study, it is best if the teacher in included as aresearcher.
and synthesis of results and conclusions
It isclear that students respond to mathematical journaling positively.The various studies show that language and writing skills come inhandy in expanding the mathematic skills of students.
and synthesis of criticisms
Themost critical issue about mathematical journaling is there is noclear indication whether teachers should mark these journals orwhether they count towards class grades. This can be tricky whenstudents are left alone to self-supervise on journaling matters. Thisis because some students have low self-supervision capabilities andrequire guidance and close supervision.
III. CLASSROOM DISCUSSION TO DEVELOP MATHEMATICAL THINKING What it is
Classroomdiscussions are teacher-student conversations pertaining to a givensubject. The discussions can be formal or informal. Informaldiscussions provide a more relaxed environment to pass ideas. Thesediscussions can happen orally in class, in groups, or online throughclass blogs and chat rooms.
Benefits of discussion
Classdiscussions promote language skills among students and also developmathematical insights at class levels. In most cases, thesediscussions are guided by the current topic and the teacher. Otherprompts can be assignments or requirements to form discussion groups.Liljedahl (2007) says that the nature of discussions depend on theeducation and teaching culture in a country. In western countries,the teacher education relationship is interactive while in someArabic countries, the relationship is master-subordinate.
Exemplars of developing mathematical thinking Article 1
One ofthe many studies on class discussions and conversation was conductedby Temple and Doerr (2012). This study sought to identify strategiesthat teachers use in interacting with students to promotecontent-rich classroom discussions in mathematics register lesson.This was in recognition of the role of teachers or assumption thatteachers were the initiators of class discussions.
Objectives or hypotheses
Thestudy relied on the one class of tenth-graders in one high school inRomania. There were 24 participants (14-boys 10-girls) from mixedsocio-economic backgrounds. The students scored 6-9 points out of 10in a mathematics test taken countrywide by all eight graders. Theteacher had over 25 years of experience teaching mathematics fromgrade five to college. She had handled the same class in the previousyear which is standard practice in Romania.
Thestudy used observational instruments to collect data. A total ofthirty lessons each lasting 50-minutes were observed, recorded ontape and transcribed for analysis. The transcriptions resulted into412 pages of text which were analyzed. Further interviews were heldwith the teachers at the beginning of the study and after.
Intotal of the lessons observed the research settled on one lesson. Inthe lesson, the class teacher was observed in action as the classdiscussed a formal definition of the Cartesian plane.
Thestudy showed that the interactional strategies used in class variedfrom one lesson episode to the other. The strategy was determined bythe goals such as recall of prior knowledge or practice talking aboutlearned concepts. Four strategies were observable, funneling,leading, focusing and probing.
Thecombination of teacher questions, correcting of feedback and responsewith correction using the accepted grammar patterning facilitatedbetter understanding of the mathematical register. Throughcorrections and engaging students in discussions, the teacher wasable to refine students’ symbolic expression of new concepts tomathematically acceptable terms.
Thestudy provides numerous explanations and examples obtained from thestudy results. This makes the study very easy to follow with conceptsclearly explained. The use of simple language makes the paper easy tofollow and understand.
objectives or hypotheses
Brownand Hirst (2007) carried out a study to investigate how talk wasutilized in an elementary mathematics classroom. Their study titled“Developing an Understanding of the mediating Role of Talk in theElementary Mathematics Classroom” sought to find out the whether asociocultural approach to employing different forms of talks in classcould enrich the learning and teaching of mathematics. The studyhypothesized that different forms of talk had different impact onstudents.
Thestudy used a teaching experiment relying on a sample of eightelementary teachers (5 female and 3 male) teaching in grades k1, k2,k4, and k7 together with their respective classes. The research tookplace in two schools located in a working and middle classneighborhood suburb in one of the cities in Queensland Australia. Theteachers experience in teaching mathematics ranged from one year to20 years. The researcher, who happened to be one of the teachers, hadover 10 years’ experience in using the sociocultural approach whileanother teacher had just one year’s experience. Consequently, inorder to participate in the study, all the other teachers weretrained on the sociocultural approach in order to apply it class.Comparisons were made between class A (grade 7 22 students) taughtby a teacher newly trained on sociocultural approach and class B(grade 7 24 students) taught by the researcher/teacher experiencedin the sociocultural approach.
Thestudy took place over a year. Mathematics class proceedings werevideotaped by a research assistant twice within the year andobserved. Further regular anecdotal observations were made onteacher-student and student-student interactions during classes.Teachers and students were asked to keep reflective journals of classproceedings. Additional interviews were held at the end of the year.They involved engaging the teachers in stimulated recall interviewsby watching their recorded classes and answering a set of questions.
Thetrained teachers were required to use the learned socioculturalapproach in teaching mathematics in their respective classes. Theoutcomes in the grade 7 class taught by the researcher who was theexperienced teacher was compared and contrasted to the outcomes ofanother grade 7 class taught by one of the newly trained teachers.
Thevideo recordings were transcribed and thematic analysis used. Teacherand student journals were also analyzed. Results in class A showedthat the inexperienced teacher successfully applied the socioculturalapproach in one of the interactions with a student named Bernice. Thestudent was afforded the opportunity to include her voice in solvinga mathematical problem rather than associating the solution with theteacher’s voice. In class B, the experienced teacher interactedwith students in a more advanced sociocultural approach. Ininteracting with the Julie, Jackie and other students, the teacherengaged the students in discursive negotiations to reach a consensusrather than calling for submission from students. Contextualdiscussions show that students were more responsive to candiddiscussions rather than correct false contexts in solving questions.
Teachersneed to be flexible to use a wide variety of discourse formats intheir class discussions,. Engaging students in candid discussion gavethe students a sense of empowerment and inclusion and respect fortheir knowledge in mathematics.
Theused deep analysis to analyze data collected. However, two mainweaknesses of the study are that the specific outcomes in terms ofimproved performance in mathematics could not be ascertained.Secondly, the study did not use a control group to ascertain that thesociocultural approach was being used by teachers by default and notjust as a strategic option.
objectives or hypotheses
Thearticle “Extending students’ mathematical thinking duringwhole-group discussions” by Cengiz, Kline and Grant (2011) assesseshow teachers` knowledge in mathematics and the instructional actionscan be incorporated in whole group discussions by stimulating morecomplex ideas for the benefit of students. The study hypothesizedthat teacher’s involvement extends students capabilities inmathematical thinking. The study hypothesized that while thisactivity was beneficial to students, it was faced by numerouschallenges. The main assumption for the study was that the maininstruction action is oral.
Theparticipants chosen for the study were six experienced mathematicsteachers teaching in grades 1-4. Experience ranged from 8-12 years.The teachers had undergone professional training on mathematicalknowledge teaching.
Theteachers were observed as they interacted with the students duringclass whole group discussions. Their instructional skill and abilityto stimulate extended thinking in the students was of interest. Thelessons were videotaped for analysis. The teachers were interviewedbefore and after the lessons were conducted.
Aninvestigations approach was used in the lessons. This approach callsfor teachers to provide great support to students to stimulate theirthinking and use their thoughts as the basis of teaching. Theresearch was interested in the fitting together of teachers’knowledge in mathematics and their instructional actions which weremainly oral. The teachers in 1st, 2nd and 3rdgrades were observed as they taught the concept of addition andsubtraction. Observation for the 4th grade teacher tookplace during a multiplication and division lesson. The observationswere made in 2-5 lessons.
Episodesin lessons during which teachers provided more than mathematicalsolution and reasoning in form encouraging mathematical reflectionwere noted. Statements related to mathematical concepts were notedand grouped into two statements that basically address themathematical content and statements that basically addressed teachingand learning mathematical knowledge. The study showed that theteaching and learning aspect which included adapted stories toexplain concepts and stimulated better thinking. Furthermore,eliciting action also proved helpful. It encouraged students toopenly talk and express their thinking about mathematical activity.Some of the effective actions were inviting students to share theirsolution methods.
Mathematicalstatements that go beyond the content and address the teaching andlearning aspect of it have more impact in stimulating and extendingthe thinking of students. How teachers talk about mathematicsinfluence how students think about it.
Thestudy carried out a primary research whose findings are morereliable. However, the observation took place over relatively shortperiod of 2-5 lessons. Given that students have different leaningcapabilities, observation should take longer.
Summary of developing mathematical thinking through discussion
Mathematicalthinking is critical in demystifying mathematics and creating anenabling environment for free flow of ideas from teacher to studentand student to student. Teachers should adapt a social cultural andeducative approach that encourages students to think in an extendedabout mathematics and apply the knowledge learnt in class.
IV. QUESTIONING AND SCAFFOLDING STUDENTS’ THINKING What it is—definitions and descriptions
Scaffolding is one of the identified teaching strategies in learning.It can be used in mathematics and other subjects. Wood, Bruner andRoss defined it as “a process that enables a child or novice tosolve a problem, carry out a task or achieve a goal which would bebeyond his [or her] unassisted efforts” (McCosker & Diezmann,2009. p. 27).
benefits of scaffolding student thinking
One ofthe key benefits of scaffolding is that it engages students in theearning process. Consequently, the strategy is not suited for thetransmission method of teaching (McCosker & Diezmann, 2009). Thestrategy has also been observed to motivate and challenge students todo better through emulation. With teachers demonstrating mathematicalsolutions, students are likely to be challenged to solve the same(ibid).
Exemplars of scaffolding student thinking with positive outcomesArticle 1
objectives or hypotheses
Thearticle titled “Teacher questioning to elicit students’mathematical thinking in elementary school classrooms” by Franke etal. (2009) seeks to find out how teacher questions shape studentsthinking.
Thenstudy used a sample from a large scale study previously carried outby Jacobs and colleagues (2007). The sample used comprised on threeteachers (two in 2nd grade and one in 3rdgrade) and their respective classes in a large urban school in aSouthern California district. The teachers were from the same school,taught in classrooms with similar structures but posted differentresults in term of students’ achievements in algebraic thinking.The school had a history of poor performance, high teacher turnoverand unaccredited teachers. The schools majority comprised Hispanicsand 99% of students were classified as minority group.
Theteachers were observed and videotaped while in action in theirrespective classrooms twice in one week. Focus was on communicationbetween teachers and students with student responses givenprecedence.
Teachersengaged students in mathematical work revolving around two algebraicquestions. The teacher posed the question to the student with studentdiscussions the possible solutions amongst themselves in smallergroups. The teacher would then take over the discussion involving thewhole class.
Resultsindicated that teachers asked four types of questions, general,specific, sequence of probing specific questions and leadingquestions. The general questions were mainly unrelated to thestudent’s response. Specific questions sought clearer clarificationfrom student responses. Probing sequence questions comprised of morethan two related questions seeking specific answers from students’responses. Leading questions sought to guide students to particularanswers and explanations. In 98% of cases, teachers asked students toexplain their thinking. In 73% teachers asked students to explainfurther. In 97% of cases, students elaborated their answers further.There were significant differences in student elaboration dependingin the questioning style used.
Teachershave the capacity to direct students thinking through questions. Thequestions should be formulated in the correct manner to bring out thecorrect type of thinking. The results also indicate that the type ofanswers could vary with the same question depending on how thequestion was frame.
Thestudy categorized student responses in a very convenient which iseasy to analyze. However, the study did not provide samples ofquestions that teacher asked even after providing the algebraicquestions that the teachers used. Nonetheless, the article usessimple language and is very easy to understand.
objectives or hypotheses
Thearticle “Using questioning to stimulate mathematical thinking” byWay (2008) seeks to show how scaffolding shapes thinking inmathematics. The study hypothesized that most teachers ask questionsthat are directed towards memorizing facts which is not appropriate.
The article relied on secondary research. The research used a totalof 10 previous studies to collect data from these study findings.
There are no discernible instruments used given that the research wasa secondary one. Furthermore, none of the secondary sources used isdiscussed in detail.
The author assessed a number of studies on use of questioning inmathematics to gain better understanding of questioning methods tostimulate and guide thinking.
There are four categories of questions starter questions, questionsto stimulate mathematical thinking, assessment questions and finaldiscussion questions. The questions are grouped into seven levelsnamely memory, translation, interpretation, application, analysis,synthesis and evaluation. The Bloom’s six levels of thinkingindicate that regular and talented students should be handleddifferently in terms of questions and scaffolding method. For regularstudents, most time in spent on imparting knowledge and least onevaluation. On the other hand, most time for talented students shouldbe spent on evaluation and the least in imparting knowledge.
This means that the questions directed to regular students andtalented students should be similar at the very first stages withchange being gradual. Where questions are used for scaffolding, alarge percentage of questions to regular should emphasize onimparting knowledge while a higher percentage of questions totalented students should emphasize on evaluation.
The article presents very practical strategies in using questioningfor scaffolding purposes and guided thinking of students in class.The article also makes a very important between methods of handlingtalented students and regular students which is not covered by noneother article used in this study. However, the article does presentand primary research findings testing the validity of the suggestedmethods.
Objectives and hypotheses
The article “Scaffolding practices that enhance mathematicslearning” by Anghileri (2006) seeks to characterize severalteaching approaches that qualify as scaffolding in teachingmathematics in elementary schools based on the originalclassification of scaffolding by Wood, Bruner and Ross (1976). Thestudy hypothesizes that teaching strategies as recommendedscaffolding by Wood, Bruner and Ross (1976) have changed over theyears.
The research was secondary in nature meaning there were no actualparticipants in the study. However, secondary studies used in thestudy collected data from elementary school children in differentgrades among them children aged 4-6 years learning geometry.
The study collected secondary data from a number of primary studiescarried on varying populations of learners studying mathematics.
The author used articles that were based on the original Wood, Brunerand Ross (1976) model. The articles fall within two broad categoriesof scaffolding, funneling and focusing. The funneling option usesquestion to scaffold students to a predetermined solution orprocedure. The focusing procedure draws attention of students tocritical aspects of a problem and then leaving the students to solvethe task.
There are three main levels of scaffolding observed rom the variousstudies. Level one is environmental provision characterized byartifacts, peer collaboration, classroom organization and structuredtasks sequencing and emotive feedback. Level two is calledexplaining, reviewing and restructuring. Reviewing involves promptingand probing, students explaining and lastly interpreting students’actions. Restructuring involves provide meaningful contest orrephrasing student talk and then negotiating means. The third levelis called developing conceptual framework and is dominated bydeveloping of representational tools. This is interdependent to twoother activities namely making connections and generating conceptualdiscourse.
Scaffolding has evolved over the years. The number and range ofstyles that teachers can employ to scaffold students in a manner thatpromotes independence of students have increased. Some strategiessuch as touching and probing which were largely viewed as preserve ofparents are used.
The study uses a comprehensive approach and easy language. It detailsscaffolding tasks that can be easily applied regular teachers withoutneed for complex training. On the other hand, the study does notassess the efficacy of some of the methods.
Summary of scaffolding student thinking research
overall summary and synthesis of benefits
Scaffolding as a teaching strategy ensures that the class is focused to a particular objective.
Scaffolding eliminates anxiety among students caused by lack of guidance in learning new concepts
Scaffolding provides instance responses to teachers on their teaching strategy though student responses on guiding questions.
summary and synthesis of instruments
It is notable that very few studies have been conducted to assess theefficacy of scaffolding n teaching. However, it is clear thatscaffolding achieves better results when used together withquestioning to guide students.
summary and synthesis of interventions
The change in teaching methods and the approach used by teaches isthe most common intervention. The intervention must be able toidentify the changes in teachers’ approaches and students’responses as a result of using scaffolding. Only then can the effectof scaffolding be assessed.
summary and synthesis of results and conclusions
Teachers are free to improvise on scaffolding methods. The basic ideabehind these methods however must be to funnel students withquestions and guide them and provide support to achieve apredetermined outcome in the case of mathematics solution. Thearticles reviewed also indicate that use of scaffolding varydepending on the students skills and abilities. Regular studentsshould spend lesser time on evaluation and more on learning theconcepts. Alternatively, talented students should spend more time onevaluation as they can understand concepts more quickly.
summary and synthesis of criticisms
There are very few studies that have been carried out on a wide scaleto test and verify the efficacy of scaffolding and questioning tofunnel and offer guidance to students. It is also evident that manyteachers misunderstand the aspect of questioning ion mathematics bypushing towards memorizing of facts such as multiplication tables.None of the studies factors in teacher miscomprehension ofquestioning.
A. implicationfor using mathematical journals
Teacher training institutes and education will focus more onincorporating mathematical journaling more in their curriculum. Thiswill give teacher better skills in employing various strategies toemploy mathematical journaling and other forms of writing forstudents. It is also expected that language teaching will be broughtcloser both in teaching curriculum and in actual teaching. Continueduse of writing in mathematics will shift the perceived value andimportance attached to different subjects amongst the public in thelong run.
B. implicationfor deep mathematical discussion to promote understanding
The study results on the role of discussing mathematics willencouraged a more relaxed environment in the mathematics lessons.Teacher will realize that they need to use a more relaxed approach inteaching mathematics and demystify mathematics as the subject thatrequires ultimate concentration compared to other subjects. Bytalking about it, it will extend the students thinking of mathematicsbeyond solving problems and memorizing formulas.
C.implication #3 for classroom practice (scaffolding student thinkingfor support)
The implication will be that there will more advanced studies toassess the impact of individual scaffolding strategies used bystudents. While there are many studies suggesting changing strategiesof scaffolding and questioning students, there are no primary studiesto support such. Teachers will be therefore more encouraged toexperiment with the various strategies in a bid to improveperformance in mathematics.
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