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.

ConceptualUnderstanding

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.

ProceduralFluency

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.

References

NAEP(2003). *WhatDoes the NAEP Mathematics Assessment Measure? *Onlineat <<nces.ed.gov/nationsreportcard/mathematics/abilities.asp>>on 19^{th}October, 2015.