CHAPTER 1
Fostering the Development of Numeracy
Selected Challenges
Numeracy is much more than performing operations with numerals. Numeracy canbe defined as the combination of mathematical knowledge, problem solving andcommunication skills required by all persons to function successfully within ourtechnological world. Numerate persons can make sense of mathematical ideas thatare part of everyday experiences and they possess characteristics that are favourablefor lifelong learning about mathematics.
Fostering the development of numeracy or mathematical literacy is the major general goal ofthe new mathematics curriculum. Implementing this curriculum presents challenges that arerelated to:
• Areas of sense making.
• Aspects of cognition.
• Characteristics in students favourable to the learning of mathematics.
Translating the critical components of the mathematics curriculum into action requiresspecial teaching strategies and instructional settings. Assessment strategies arerequired that reflect the learning outcomes related to major goals of the curriculum.Reports to parents need to include information about indicators of sense making,aspects of cognition as well as characteristics favourable to the learning ofmathematics.
A variety of reasons exists for the fact that students arriving in the intermediate gradeswill be at different stages as far as having reached the learning outcomes from theprimary grades is concerned. Diagnostic settings and strategies will be required forsome of these students before effective Individual Educational Plans – IEPs – can bedeveloped and presented.
Since the critical components and the goals of the new curriculum are quite differentfrom the mathematics teaching and learning experiences that teachers and parentshave had, new information needs to be shared, analyzed and assimilated. In orderfor parents to be able to reinforce and supplement what students experience in themathematics classroom, important information needs to be made available to them.
The Purpose and the Parts of the Book
The main purpose of this book is to identify the important components and learningoutcomes of the curriculum for the intermediate grades. Suggestions are made for teachingstrategies and assessment techniques deemed appropriate for reaching the key goalsand major learning outcomes. The suggestions in the book include ideas for:
• Teaching strategies, classroom settings and sample questioning techniques.
• Types of activities, problems and appropriate practice.
• Sample assessment tasks and suggestions for reporting to parents.
• Diagnostic tasks, settings and strategies.
• Reflection or discussion.
Whenever appropriate, discussions will begin with the identification of important prerequisiteideas, procedures and skills. Examples from these discussions are suggestiveof introductory teaching-learning settings and they can also be used for diagnosticpurposes. The responses that are elicited during diagnostic interviews can be used forplanning effective intervention – Individual Educational Plans (IEPs).
The intent of the questions that are included at the end of some sections and at the endof each chapter is to provide ideas for reflection or for initiating discussions.
Theoretical comments will be kept to a minimum. Brief reference will be made to generalguidelines that research provides with respect to teaching, learning and assessment.Suggestions will be made for accommodating selected types of responses from students.
The suggestions that are made and the examples that are cited in the book are of apractical nature since they are based on:
• Experiences with students in classrooms.
• Observations in classrooms of teachers interacting with students.
• Action research conducted with teachers and students in classrooms.
• Responses collected from students during diagnostic interviews.
• Transcripts of diagnostic interviews with students and conversationswith adults.
Challenges Related to Curriculum Content
Aspects of Sense Making and Problem Solving
One of the key goals includes the maintaining and fostering of all aspects of sense makingin mathematics. These aspects include:
• Number sense.
• Spatial sense.
• Measurement sense
• Statistical sense.
• Sense of relationships.
• Developing and applying new mathematical knowledge throughproblem solving.
Problem solving is one of the major goals of teaching students about mathematics.The curriculum includes the statement that, 'learning through problem solving should bethe focus of mathematics at all grade levels' (p.8). It is in these through or via problemsolving settings that students develop their own problem solving strategies. New problemsolving strategies may be encountered when students are given opportunities to discussand compare strategies. An awareness of new strategies may lead to attempting these inthe future.
The key to creating a through problem solving setting lies in how tasks are presentedand how questions are phrased and posed. The onus is put on the students who arerequested to use what they know and try to come up with solutions or suggestions fora solution. The following are examples of types of questions and requests that invitestudents to invent which is one component of a balanced and effective mathematicsprogram:
How would you or could you ...?
Try to think of at least two ways to ...
The ability to respond to these types of requests is dependent upon one or more pre- orco-requisites. These can include: number sense; spatial sense; ability to think flexibly;ability to visualize; ability to generalize; ability to connect; ability to estimate; high self-esteem;willingness to take risks. The accommodation and development of these aspectsof cognition and these favourable characteristics has to be a component of ongoinginstructional settings in the mathematics classroom.
A request that includes, 'Try to think of at least two different ways to ...' not only serves toaccommodate individual differences, but illustrates the important idea that it is better tosolve a problem in many different ways than to solve many problems in the same way.This idea and more is beautifully illustrated by Calandra's story who was asked to be areferee on the grading of a student's response to an examination question on a physicsexam.
The student had answered a question about determining the height of a tall building withthe aid of a barometer with the suggestion to take the barometer to the top of the building,attaching a long rope to it, lowering the barometer to the street, bringing up the rope andthen measuring the length of the rope to determine the height of the building. The studentappealed his grade of zero, since according to him he had answered the questioncompletely and correctly and he claimed that he would get a perfect mark if the systemwas not set up against the student.
The appeal resulted in giving the student six minutes for another opportunity to record aresponse with the proviso that his answer should reveal some knowledge of physics.Since after five minutes he had not written anything, he was asked if he wished to give up.The response of 'no' was accompanied by the comment that he had many answers to theproblem and that he was just trying to think of the best one. In the next minute he recordeda response that made reference to dropping the barometer, timing its fall with a stopwatchand using a formula to calculate the height of the building. Almost full credit was awarded.
When the student was asked about the other responses he had in mind, he referred tomeasuring shadows and using proportions; swinging the barometer as a pendulum at thetop and the bottom and calculating different values of 'g'; and using the barometer as a unitof measurement to count the number of 'barometer units' it takes to describe the height ofthe building. His final suggestion consisted of offering the barometer to the superintendentof the building in exchange for being told the height of the building. The student did admitthat he knew the conventional answer, but that he was tired of instructors trying to teach himhow to think in a pedantic way.
Teaching through or via problem solving requires asking appropriate questions,orchestrating follow-up discussions and accommodating different types of responses.Students are given the opportunity to develop their own problem solving strategies, to thinkabout their thinking, and to become flexible in their thinking. Listening to others anddiscovering new ways of approaching or solving problems can result in advancing students'thinking. The success of orchestrating the complex instructional setting that is required forreaching these desirable results is dependent upon the important role played by a skillfulteacher.
Aspects of Cognition
The fostering of aspects of cognition related to learning about mathematics includes:
• Conceptual understanding.
• Visualization.
• Mathematical reasoning.
• Mental mathematics and estimation.
• Aspects of thinking: thinking flexibly; thinking about thinking; algebraic thinking.
Whenever possible strategies need to be considered that can advance students' thinking.
The aspects of sense making and cognition allow students to develop mental mathematicsstrategies for the basic facts and to create personal strategies for computational procedures.
Conceptual Understanding
Research shows that instructional settings can emphasize conceptual understanding withoutsacrificing skill proficiency. An emphasis on skill proficiency and meaningless mastery willnot contribute to any aspects of conceptual understanding.
Key indicators of conceptual understanding include the ability to:
• Talk and write about what has been learned in one's own words.
During conversations and diagnostic interviews it becomes apparent thatmost students and adults lack this ability. Responses by subjects torequests for explanations of computational procedures are very similar.These explanations begin with,
'First you write one number below the other. Then you start at the ones....'
Some time ago, an instructor turned entertainer made a song out of the rotechants to find answers. The opening line of his song about teachingsubtraction includes the key words of the chant familiar to most students,'You can't take ... from ..., so you go to the ... in the tens place.'
Conceptual understanding implies that procedures and reasons for theprocedures can be explained in one's own words, no matter howinadequate someone may judge these to be at the early stages of learningabout a procedure. The acquisition of this ability is dependent upon studentsbeing given many opportunities to share explanations as part of theirmathematics learning.
• Connect what has been learned to experiences outside the classroom and to othersubject areas.
Interviews yield many examples of students' inability to connect. Manystudents are unable to tell who might use the algorithmic proceduresfor multiplication or division and when and where these might beneeded. Many are unable to connect decimal and integers to anythingoutside the classroom.
• Connect what has been learned to previously learned aspects of mathematicslearning.
This ability to connect and to tell how what is being learned is differentand how it is the same from what was learned previously can facilitateunderstanding of new learning. For example, students who know howthe divisive actions are the same and how they are different from themultiplicative action will be better able to connect the actions to theirexperience.
Visualizing
The ability to visualize is an important part of key aspects of sense making and the abilityto solve problems. Fostering the development of visual thinking and the ability to formvisual images requires special types of activities and special types of questioning. Aprovision and use of concrete materials, technology and visual representations is notsufficient. The right questions and proper problems need to be posed or presented asmaterials are handled. Without high-order thinking questions and open-ended problemslittle, if anything related to the desired mathematics learning outcomes will 'travel' fromthe hand or the hands, up the arm to the brain.
During diagnostic interviews many indicators of students' inability to visualize cansurface. For example:
• Inability to visualize the numbers for given number names.
• Inability to illustrate or recognize the action of operations and the orderof the numbers in pictorial representations.
• Inability to translate equations into pictorial representations.
A teacher's role in fostering the development of visualization is essential. Appropriateactivities, problems and questions need to be considered during the planning stages.Without a conscious effort of attempting to focus on fostering visualization this importantaspect of mathematics learning will not develop.
Mathematical Reasoning
The information processing or thinking that is part of learning and doing mathematics is notvery different from most kinds of thinking that human beings do. Greenwood identifiesseven criteria of mathematical thinking (p.144):
• Everything you do in mathematics should make sense to you.
This sense making implies the presence of conceptual understandingwhich includes the ability to talk about mathematics in one's own words.Davis reports what he calls 'disaster studies' since the subjects that wereinterviewed could not talk about the mathematics they had learned. Thisdilemma can be attributed to the type of requests that are made by someteachers, tutors or parents. For example:
* Just do what I tell you to do, it will work and you will getthe correct answer.
* Just follow these rules and steps.
* Just memorize these steps and you will always get theright answer.
* Just take this short cut because it is much faster.
* Let me show you a trick that will always work.
It is very discouraging to hear university students, and that includes teachers-to-be,utter comments like, 'Don't tell me why it works, just show me how to doit' or, 'Just tell me what I need to know to pass the test.' Many students at thislevel express gratitude to the teachers who told them what to memorize andwho coached them to pass tests. It may be quite a challenge for theseteachers-to-be to create settings of sense-making once they get a classroomof their own.
• Whenever you get stuck, you should be able to use what you know to get yourselfunstuck.
Sense making is a pre-requisite for this ability. Numbers and operationsneed to make sense before students can develop and acquire the mentalmathematics strategies that will make it possible for them to getthemselves unstuck.
Students also need to be confident and willing to take cognitive risks. It isa little discouraging to observe students in classrooms and to interviewstudents who lack these characteristics. They are afraid to try somethingthat might be judged as incorrect.
* You should be able to identify errors in answers, in the use of materials, and inthinking.
Sense making and conceptual understanding can be fostered in studentsby having them look at and examine examples as well as non-examples.Non-examples can become part of demonstrations as well as appropriatepractice exercises. For example, one teacher concluded a lesson byrecording calculation procedures on the chalkboard that contained errorsand her challenge to the students was to try and find out what they thoughther brain had been doing wrong. Practice sheets can be prepared whichinclude descriptions of solution procedures that are incorrect or diagramsthat are inappropriate or labelled incorrectly.
• Whenever you do a computation you do a minimum of counting.
Perhaps the criterion could read, do something other than counting, sincecounting is not a strategy. Mental mathematics strategies and personalstrategies are based on possessing a sense of number which includesvisualizing and flexible thinking about numbers and do not depend oncounting or counting by ones. The goal should be to do computationswithout counting.
