Learningin most cases is done with intent or an end goal. This paperdiscusses the various patterns of learning that are employed acrossthe whole class. In doing the discussions, the patterns of learningwill be done relative to a number of issues including conceptualunderstanding, the procedural fluency employed and the mathematicalreasoning or even the problem solving skills that are used. Relevantexamples will be drawn from the summary presented.


Thisentails having the students gain the full understanding of thevarious mathematical concepts, operations, and relations (NAEP,2003). A case example in the summary is the evaluation criterion 1,labeled as #1. This criterion sought to establish whether studentswould use context clues to determine or clarify meaning of some ofthe unknown words or phrases like assessed in question 1a-2d. Thiscriterion was set to ensure that students have conceptualunderstanding of subject. In this case, students are to show proofthat they can recognize, label, and even generate certain examples byuse of concepts. The ability of students to demonstrate that they areable to apply models and diagrams that are varied is a classicexample of conceptual understanding. Criteria 1 and 4 of the learningas presented in the summary are evaluating the conceptualunderstanding of the students. Both the criteria test the ability ofthe students to reason in various settings that are provided to them.Reasoning and reflection as expected by both the criteria presumablyis an application of various concept definitions, relations or eventhe representations defined in the summary. A clear depiction isevident in question 1a to 5d.


Whenteaching and learning, this skill tests the ability of carrying outthe procedures in a flexible manner, accurate, efficient and in anappropriate way (NAEP, 2003). The summary presents the test of thisthrough criteria 3 where the students are expected to read, write,and model number to 999. Carrying out of these three activities mustbe in procedure form and follow sequence. The student has first todemonstrate ability to read, then write and then model numbers insome sequence say from 1 to 999. This is an area that has been wellprepared and an overwhelming 95% of students mastered the concept asevidenced in their performance in question 3a-5d.

MathematicalReasoning/Problem Solving Skills

Learningthrough mathematical reasoning or by application of problem solvingskills mainly puts focus on assisting the students to get deeper andbetter insight of the various mathematical ideas and processes. Thishappens when students are engaged in doing mathematics throughcreation, exploration and even verification of concepts (NAEP, 2003).Thus, ideally reviewing the summary, it is apparent that there iscapacity for logical thought, reflection, explanation, andjustification that is assessed through evaluation criterion 4. Inthis criteria, the deeper insight is sought when the students areasked to use identify and use word and models. They are additionallyasked to use expanded form to represent the numbers to 999.

Whenexpanding the numbers, the students have to build on earlierknowledge learnt. Therefore, the students have to do some kind ofreflection on earlier learnt knowledge. In order for the students torelate hundreds, tens and ones, they must have proper mathematicalreasoning that is coupled with mastery. In some form, the studentsare being asked to reflect on what they learnt about ones, tens, andhundreds and then use them to learn. This is an integral part thatthe evaluation tests in criterion 2. The summary depicts that 60% ofstudents have learnt it. This means that more has to be done toensure that students can comfortably relate the concepts, reflect onthem, justify, and even give a logical thought without which theywill not be able to relate tens to hundreds and ones.


NAEP(2003). WhatDoes the NAEP Mathematics Assessment Measure? Onlineat &lt& 19thOctober, 2015.

Institution Affiliation:

Howa person prefers to learn can be referred to as a &quotlearningstyle.&quot No learning style can be perceived as good or bad: themost important issue is that information is imbibed into the tutee`smind effectively. Thus, a tutor`s learning style influences histutoring style profoundly. If a tutor`s learning style clashes withthat of the tutee, trouble and frustration is bound to ensue.Identifying and defining both styles, therefore, becomes a pivotalissue in developing a successful teacher-student relationship(Montogomery and Graot, n.d.). This paper discusses the learningpatterns of 20 students emphasis will be placed on conceptualunderstanding, mathematical reasoning and procedural reasoning.


Conceptualunderstanding is perceivable as how students choose to learn, forexample, mathematics, with understanding. This type of understandingbuilds new knowledge from prior knowledge and experience (Gordon,2006). In the case study, where 20 students were assessed, someinconsistencies in conceptual understanding were noted. For example,some patterns were identified: 1/20 students missed question 1a, 3missed 1b, 1 missed 2a, 2 missed 2b, 4 missed 4a, 2 missed question2, 1 missed question 3, 2 missed questions 4 and 5, 1missed question6, 8 missed questions 7 and 9, and 8students missed question 9 on thevocabulary section of the test. The before-mentioned issues mean thata good number of students were incapable of comprehendingmathematical concepts, relations and operations. Focus should,primarily, be placed on equipping students with the requisite skillsto help them learn mathematics: helping them become mathematicallyproficient.


Thesecond learning style is procedural fluency. This learning style,also referred to as rote learning or procedural knowledge, paysattention to understanding mathematical concepts. Skills such asbeing flexible in carrying out procedures, accuracy, efficiency andappropriate when it comes to gaining and applying mathematical skillsare emphasised (Gordon, 2006). The case study reflected a strugglewith vocabulary comprehension. The majority of learners were not ableto read instructions carefully, made careless mistakes, and did notuse contextual clues to answer questions. The difference betweenprocedural and conceptual understanding is that the latterconcentrates on acquiring skills that help a student find solutionsto problems while the former concentrates on helping a student becomeproficient in applying mathematical skills.


Thelast approach is mathematical reasoning. This approach emphasises&quotpracticalness.&quot Students are helped to understand whycertain principles should be applied (Stacey, 2007). For example,understanding the rationale behind the formula for finding thesurface area of a cylinder: circumfrence×height+2×area of the base. Students are helped to think about mathematics in a different way.According to the case study, 35% of the students were able to applyevaluation criteria number 1: they used context words to derive themeaning of unknown words and phrases. 90% of the students were ableto comprehend the skills that related to evaluation criteria number2: students were able to make a relationship between hundreds, tens,and ones. 60% of the students were also able to understand evaluationcriteria number 3: they read, wrote and modelled numbers to 999.Finally, 85% of the students were able to master evaluation criterianumber 4: they identified and utilised models, words, and alsoexpanded form to make a representation of numbers to 999.


Gordon,&nbspF. (2006). What Does ConceptualUnderstanding Mean?&nbspNewYork Institute of Technology.Retrieved from

Montogomery,&nbspS., &amp Groat,&nbspL.(n.d.). STUDENTS LEARNING STYLES AND THEIR IMPLICATIONS FOR TEACHING.Retrieved from

Stacey,&nbspK. (2007). WHAT IS MATHEMATICALTHINKING AND WHY IS IT IMPORTANT?.&nbspUniversityof Melbourne, Australia. Retrievedfrom