Wenglinsky, H. (1998). Does it compute? The relationship between educational technology and student achievement in mathematics. Princeton, NJ: Educational Testing Service. Retrieved March 6, 2002, from ftp://ftp.ets.org/pub/res/technolog.pdf.

Preface: 

In this age of technological imperative, we do it simply because it can be done. Massive efforts are underway to convert traditional teaching to something that can be delivered via computer.  Measuring success has been a simple matter: count the number of computers, divide that by the number of students, and report how the ratio of computers to students has advanced-and it is always advancing. Then close the report lamenting that we don’t know much about the software being used on these computers, we don’t know how many are behind locked doors, we don’t know how many are broken, and we don’t know how many teachers really know how to use them (there are no assessments of teacher capability here). 

The Policy Information Center reported in May of l997 what we did know at that time, in Computers and Classrooms. We said then that the available data, and the research performed on it, did not tell us whether computer delivered instruction actually improved performance.  This report takes at least the first step in determining whether computer use is making a difference in mathematics, and what kind of computer use has what kind of effect, on which groups of students. 

This study uses a national database, the 1996 National Assessment of Educational Progress, and advanced analysis techniques, to isolate the effects of the computer from the myriad other factors involved in student achievement. The study was suggested to us by Education Week, and its author, Harold Wenglinsky, of the ETS Policy Information Center, has collaborated with the staff of Education Week throughout the analysis and writing phases of the report.

In addition to telling us what he found, Harold Wenglinsky also tells us what further research must be conducted to learn more, and give the nation specific guidance in its efforts to raise educational achievement through greater use of technology.

RQs

1. Does computer use make a difference in mathematics?
2. What kind of computer use has what kind of effect on which groups of students?

This report presented findings from a national study of the relationships between different uses of educational technology and various educational outcomes.  Data was drawn from the 1996 National Asssessment of Educational Progress (NAEP) in mathematics consisting of national samples of 6,227 4th graders and 7,146 8th graders.  The NAEP consists of tests in various subject areas administered to national samples of students, usually every 2 years.  It is used to measure trends in student performance overtime and between subgroups.  It also includes questionairres given to students, their teachers, and principals to put test scores into their educational context.  In 1996, it was the first time NAEP included questions on technology.  NAEP includes information on: frequency of computer use for math in school, access to computers and frequency of computer use in the home, PD of math teachers in computer use, and kinds of instructional uses of computers by mathematics teachers and their students(higher-order v. lower order).  Those four categories serve as the indicators for the organization of technology use and are related to 2 educational outcomes (academic achievement in math and social environment), along with each other.

The study used advanced analysis techniques and specifically examined the subgroups of ethnicity, gender, region, public v. private, economic status, and community status (rural, urban, etc.).  The study found that essentially technology could matter, but it depends on how the technology is used.  It also found that there is great inequity in how computers are used.  Specifically, the study found that (for 8th graders):

• Black students were less likely to be exposed to higher-order use of computers and more likely to be exposed to lower-order uses than White students.  Similarly, so were poor, urban, and rural students as opposed to non-poor, suburban students.
• Urban and rural teachers were less likely to have mathematics teachers who had received PD in technology over the last 5 years than suburban students.  Similarly, the same goes for poor vs. non-poor.
• Black students were less likely to have access to a home computer than White students, and this was similar for poor and urban students as well.

Specifically regarding academic achievement, the study found that:

• Teacher’s PD in technology and the use of computers to teach higher-order thinking skills were positively related to academic achievement.
• The frequency of home computer use is positively related to academic achievement.
• The use of computers to teach lower-order thinking skills was negatively related to academic achievement.
• The frequency of computer use was negatively related to acedemic achievement.  “DON’T JUST USE TECH FOR TECH SAKE!”

It is important to note some methods of this study.  First, “frequency of computer use” is based on student reports and ranges from “never of hradly ever’ to “almost everday”, with the option of “there is no computer at home” for the home use questions.  PD is based upon asking teachers whether they have received such development in technology, specifically computers, in the last 5 years.  For 8th graders, high-order use is measured as “simulations an applications” and lower-order uses are measured as ”drill and practice”.  Three characterisitcs of students and schools are taken into account: SES, class size, and teacher characteristics.   This means that all relationships between technology and educational outcomes reported represent the value added by technology for comparable groups.

It is apparent from this study that there are clearly equity issues within our educational system that include the use of educational technology.  It appears that there exist somewhat of a socioeconomic digital divide.  Disadvantaged groups seem to lag behind in the aspects of technology that do affect educational outcomes.  The answer starts with education at the top.  Students with teachers who have PD show higher levels of achievement.  Policymakers in all sectors involved with education should focus on adminstering educational technology in an equitable fashion.  This includes not just the equipment itself, but educating teachers in order to educate students.  Teachers need good PD (including applicable and ability to attend) in the areas of higher-order thinking and technology use and thus incorporating it into the classroom. 

Some limitations of this study are:

• Data was collected at a single point in time, and thus certain subgroups were not followed over a period of time to track results.  Additionally, aspects of technology studied here occurred at the ssame time as the educational outcomes of interest.  And so it may be that high-acieving students are more liely to use technology in certain ways rather than that these uses of technology promote high levels of academic achievement.
• The study does not take into account the tendency of teachers to teach in certain ways or detailed measures of teacher practices.  Thus it may be that computers are more of a medium conducive to high academic achievement. 
• The study does not take into account information about state technology policies, which could have a bearing on effectiveness.  It has been noted that states on left on their own in how to implement educational technology.

McCoy, L., P., (1996). Computer-based mathematics learning. Journal of Research on Computing in Education, 28(4), 438-460. CARET summary.

Abstract:  The research on computer-based mathematics learning was examined in the context of constructivist learning and the National Council of Teachers of Mathematics Standards.  The reviewed studies examined the effectiveness of computer software in three key categories of programming, computer-assisted instruction, and tools.  Recommendations emphasized the need for further study of instructional design that best utilizes computers and use of the Internet in mathematics.

RQs

1. How can technology influence student academic performance?
2. How can technology develop higher order thinking and problem solving?
3. How can technology be infused into curriculum and instruction effectively?

Notes:

This article is basically a culmination in research on computer-based mathematics learning beginning around 1988 and through the mid 1990’s.  It focused on three areas: programming, computer-assisted instruction, and tools all with a basis rooted in constructivst learning, as advocated by the NCTM.  The author refers to the progamming environment as a mathematical laboratory where students can manipulate mathematical ideas as they create their own concepts.  In this environment, logic is established and students may practice and develop their reasoning skills.  In the computer-assisted instruction environment (CAI) computers are used in assistive modes such as drill and practice, tutorials, or simulations.  Simulations model the real world in some manner and include microworlds, which create a model environment for students to actively experience mathematics (in line with constructivist theory).  Tools refers to software that assists the user in performing some mathematical function.  The culmination of all this research indicates that computer-based materials are effective in facilitating mathematics achievement.  This is particularly true in Logo programming (note: Scratch’s grandfather), CAI microworlds, and algebra and geometry tool software.  However, the author notes that instructional materials and methods must be evaluated in terms of their usefluness and overall goals.  The author also stresses that useful research needs to occur in the area of instructional design, as a key recurring conclusion of the studies indicate the importance of teacher, designer, and researcher input in structuring computer-based activities such that the students can understand procedural and conceptual knowledge properly.  (Technology counselor?????)  Specific research findings in the 3 areas of computer-based mathematics learning are discussed in a bit more detail below.

In the “programming” mode, students work with reasoning to “teach” the computer.  In doing so, they are creating mathematics and the computer is providing immediate feedback to assist them in exploring and refining their knowledge.  Much of this research has had to do with LOGO, but there is some use of BASIC as well.  The research focuses on the effect of programming on students’ cognitive abilities, though it is generally difficult to measure problem-solving skills.  As for assessment, most of the studies used interviews on concepts and/or a post-test.  Overall, the studies showed that:

• LOGO had better results than BASIC and is most effective in learning geometry, but can also be effective in learning variables.  It is believed that the very visual nature has much to do with this.
• Research shows that specifically, learning infused with LOGO had some positive success in the topics of distance, figural creativity, angles/shapes/motion, angles/polygons, fractions, problem solving, and variables. 
• Shorter treatments had significantly higher achievement than longer treatments.
• Programming experiences provide a “mildly effective approach for teaching students cognitive skills in a classroom setting.”
• LOGO programming, especially turtle geometry at the elementary level, is an effective medium for providing mathematics experiences.
• Teacher needs to be involved in planning and overseeing the LOGO experience to ensure that students discover and understand the target concepts.  Otherwise, unintended results can occur rather easily.  As one study by Olive found, “…Students had to do well on LOGO tasks in order to do well on the geometry measure, but that some students were successful on the LOGO, but not on the geometry.”  Clements and Batista also found that while the LOGO group improved more in student achievement than control group, both groups still had misconceptions about angles and figures, though the LOGO group had more experience and were able to construct and communicate the concepts. 

CAI use of computers is a more structured use, but potentially powerful none-the-less.  In general, research showed that CAI is effective because it allows students to experiement through active manipulation within a structure of a mathematical model (microworlds, tutorials) and serves as supplements (drill) with immediate feedback and direction.  The research focuses on students’ learning of mathematical concepts, as emphasized by the NCTM standards and most assessment was done by interviews and posttests.  The author really focused on microworlds, citing their ability to to be used for exploration in a constructivist mode, yet containing a definition of certain mathematical tools, or in other words, built in varying levels of scaffolding.  Many micorworlds were often created in LOGO.  (This may not be as true now…)  However, they were all different in how they are created and used, which made it difficult to make generalizations and emphazised the need to evaluate materials in their usefulness and overall goals.  For instance, Blocks was created by Thompson and used to help students understand the relationship between blocks and numerals (symbolic/iconic).  His research indicated that it did not help students learn the relationship and that they could not unlearn their incorrect algorithms from whole numbers.  On the other hand, Dots and Rules (Zehavi) was deemed to be effective in helping students in understaning graphing.    Four studies found specific microworlds to be effective in helping students learn mathematical concepts.  Those studies included the use of Motions, which focused on geometry and was evaluated using interviews and a delayed posttest.  Other microworlds include one by Thompson and Wang for working with coordinate Cartesian graphing, one by Edwards, working with tranformational geometry, and Dots and Rules.  Five major features of a microworld were identified: changes in problem situations, immediate feedback, motivation, flexibility and dynamic connections between symbols and graphics, and integration of spatial and graphic representations.  This appears to be a common theme in much research regarding technology based learning.  One negative from the microworlds is that a visual solution can be acheived without conceptual understanding.  Thus, explicit connections must be established in order to link procedural knowledge and conceptual understanding (prinicple #2).   Teachers must facilitate guided discovery, a theme that consistently appears in the 3 categories of research here as well as outside these studies.

Research of other types of CAI (drills, tutorials) shows that, potentially,  problem solving skills can be improved through the use of software, but again different software had differing results.  It warrants the mention that further examination has to be done on the differential effects of mathematics software on different students.  One study found that CAI increased the gap between low and high achieving students.  So does software need to be better geared towards specific learners? It also warrants mention that groups and pairs had higher achievement in CAI than individual students.  This aligns with the constructivist learning theme.

Mathematics tools are advocated highly by NCTM for accomplishing routine calculations so that students can focus on conceptual understanding.    They are also useful in the constructivist mode for providing models of different parallel representations to help students build on conceptual connections.  The mathematical functions performed most effectively by computer tools are algebra (graphing, symbol manipulation) and geometry (construction, visualization of figures).  Most math educators would agree that a graphing utility is an absolute necessity in algebra or calculus.  Graphing software is also important because it facilitates work with real-world problems (ex: statistics).  Some common software applications that are used are: Maple, Derive, Mathematica, Theorist, MathCad (in algebra they promote experimentation and concept building by allowing a linked representation of algebraic, tabular, and graphic models so that any change is reflected in all representations), Geometric Supposer, Geometer’s Sketchpad, Cabri:The Interactive Geometry Notebook (in geometry they provide a variety of geometric constructions and measures).   Overall research showed that results varied among the use of the previously named software applications, with a general increase in conceptual skills, but no difference in most overall math skills.  This has also been a common theme in research – increase in problem-solving and concepts, but no difference in the computational and manipulation skills to get the right answer!

Boster, F. J., Boster, L. J., Yun, J. A. Strom, R. (2005). Evaluation of New Century Education Software 7th Grade Mathematics Academic Year 2004 – 2005 at Grant Joint Union High School District. Cometrika, Inc.

Executive Summary:

•  An experiment was conducted during the 2004-2005 academic year in which 454 7th Grade students participated in a mathematics experiment in order to assess the effectiveness of the New Century Education Integrated Instructional System.  Experimental group students received exposure to the New Century Education
  system; control group students did not.
• Student performance was assessed by comparing 2004 and 2005 CST and CAT6 scores for experimental group and control group students.
• This was part of a larger controlled study that assessed 1,293 7th and 9th Grade students who participated either in a mathematics experiment, a reading experiment, or both.
• Mathematics results indicated a strong effect for 7th grade. The 7th grade effect indicated that students exposed to the New Century Education Integrated
  Instructional System outperformed those who were not exposed to New Century.
• Limitations of the experiment include relatively modest sample sizes and relatively large attrition.

———————————————————————————————

• RQs
  1. Does New Century Education Software enhance student performace in mathematics?
  2. If so, what is it about the software that contributes to such enhancements?

This research article is based on a study of 454 7th grade math students in a large, urban, diverse school district in Northern California.  It involved 6 schools.  The students involved in teh study were “stragetic students” in that they were performing 1 – 2 levels below grade level.  In the sample, 51.3% of the students were male, 65.2% predominantly spoke English at home (18.3% spoke Spanish), 30.4% White, 30.8% Hispanic, 18.9% Black, 15.2% Asian/Pacific Islander.  The students were assigned randomly to a control group and an experimental group (the group using the New Century Education software).  Students in the control group received basically the same instruction, but without the software.  Students in the experimental group were expected to use the software for at least 90 minutes per week.  Teachers of the experimental group were expected to integrate the technology into their lessons.  Pre and post CST and CAT6 tests were used for evaluation purposes.  TOT methods within the software were used to confirm number of activities that students completed, number of minutes on task, and number of days logged on.

What is the New Century Education Software?  It is instructional software that is tailored to individual student needs.  It is based on, and further researching, the notion that instruction based on individual needs will help students learn more effectively and that technology can help create that kind of taylored instruction.  It promotes less-traditional methods of teaching and helps teachers use technology to engage different types of learners in mathematical instruction.  (He also cites his other work on research video-streaming and improved math performace).  In a nutshell, it provides reinforcement, immediate feedback, strong audio content,  and personalization features using animations, graphics, concrete examples and audio that helps learners engage intead of just reading information.  The software is created for the “norm” students, not special needs or LEP students.  It is created for those average students that continue to fall behind as their education continues, relative to other students.

The results of the study indicate that the post-test scores for the experimental group tended to be substantially higher, suggesting that the software had an impact and improved mathematics achievement on important dimensions of the CST and CAT 6 examinations.   Students in the experimental group outperformed those in the control group.  (The study included standard deviations and other calculations to support the claim for significant and substantial effects.)  Unfortunately, the author did not go into detail on specific areas or dimensions of the tests that improved (or didn’t improve).  The author notes some significant limitations of the study, namely the small sample size – that got even smaller when the scores were removed for LEP students and students who weren’t able to complete both tests, and the limitation to one geographical area in only 1 school district.  A good amount of attrition occurred.  It lacked the size to to detect small but important effects, and point parameters were not as stable as they could be or one would like them to be.  He also noted that the treatment was conservative – only 90 minutes per week – and might have had a more profound effect if used more frequently. 

I liked this research study.  While it had its issues, it was very straightforward.  He also has several other similar research projects, and some continuing ones for this effect that could help strengthen his position.  His theme is recurrent in regards to individual student needs, but does not take into account any relationships between the actual teachers and technology.  There is also no detail on how the teachers integrated this technology or the methods used.

Cuban, L., Kirkpatrick, H., & Peck, C. (2001). High access and low use of technology in high school classrooms: Explaining an apparent paradox. American Educational Research Journal, 38(4), 813-834.

Abstract:  Most policy makers, corporate executives, practitioners, and parents assume that wiring schools, buying hardware and software, and distributing the equipment throughout will lead to abundant classroom use by teachers and students and improved teaching and learning. This article examines these assumptions in two high schools located in the heart of technological prog- ress, Northern California’s Silicon Valley. Our qualitative methodology in- cluded interviews with teachers, students, and administrators, classroom observations, review of school documents, and surveys of both teachers and students in the two high schools. We found that access to equipment and software seldom led to widespread teacher and student use. Most teachers were occasional users or nonusers. When they used computers for classroom work, more often than not their use sustained rather than altered existing patterns of teaching practice. We offer two interrelated explanationsfor these challenges to the dominant assumptions that guide present technological policy making.

• RQs
  1. With abundant access to information technologies, did the national patterns of infrequent and limited teacher usage of computers emerge?  If so, why?
  2. Did teachers who used computers in their classrooms for instruction typically maintain existing practices?  If so, why?
• Units = 2 high schools located in the hot-bed of technology – The Silicon Valley
  • Schools – both similar
    • Modestly affluent area
    • Racially and socioeconomically diverse student populations
    • Per pupil expenditure of ~ $5200
    • Abundant in technology resources based on national standards (from a hardware/software/Internet standpoint though it doesn’t mention too many specifics)
      • 4 – 5 students per computer (NA is 6)
      • 17 – 22 computers in a classroom (not every classroom, NA is 17)
      • 8 – 12 computers in a lab (NA is 21)
      • 64 – 80% Internet connectivity in classroom (NA is 44%)
      • 80 – 100% Internet connectivity in lab (NA is 54 -55%)
      • 100% of teachers have email (NA is 39%)
  • Students
    • Middle-range SAT scores
    • Aspire to higher education
  • Teachers
    • Mostly White
    • Some Latino
• Treatment = observational research study, no treatment applied
• Observations (methods)
  • Interviewed 21 teachers and 26 students that volunteered to be part of the study
  • Shadowed 12 students and 11 teachers in both schools through and entire school day.
  • Surveyed 1/4 of the student body of one school and 1/3 of the other.
  • Reviewed teacher sign-up data for media centers and computer labs.
  • Examined accredidation reports, proposals for launching reforms, grants seeking tech funds, and newspaper articles written about the schools.
  • Results of surveys and
• Settings
  • Student outcomes
    • Most students had access to computers at home and that use was frequent and spanned many applications.
    • Interviews of 33 student volunteers reported serious to occassional use of computers and other technologies in English and social studies classes and in tech-heavy classes for word processing, Internet searches, and completing projects.  For the majority of their academic classes, they reported liitle or no use of computers, but some use of videos, TV programs, and overhead projectors.
    • Based on surveys, there were students for whom technology use created computer competence for them and increased their self-confidence and motivation to do well in school, called “open door” students.  They were predominantly male and from varied ethnic backgrounds and primarliy self-taught computer users who learned from using computers at home and whose expertise far exceeded that of their classmates.  They along with “tech gods” used their technological experience and expertise to help teachers and students, which helped to ease the demand on overburdened and understaffed technology support team (which between both schools consisted of 2 people, one of whom taught 3 classes).
  • Teacher outcomes
    • Most teachers had access to computers at home and that use was frequent and spanned many applications
    • Teachers reported they largely used computers to prepare for class.
    • Teachers in both schools take their students to the media centers where they were sufficient machines to accommodate the entire class.  A few teachers (25% in one school and 32% in the other) in 3 departments (English, science, and social studies) accounted for 60 – 70% of all machine use in the media centers.  And thus, almost 2/3 to 3/4 of teachers who taught academic subjects in both schools were nonusers of the media center resources.
    • Within departments, a pattern was seen where one or two teachers were heavy users of computers, a few were occassional/rare users, and the rest were non-users.
    • Data gathered showed general lack of usage among teachers in classrooms, labs, and media centers.
    • In interviews with 21 teachers, 13 (60%) said use info tech changed their teaching practices.  They planned more efficiently, communicated with colleagues more often via email, and secured info over the Internet.  They saw students’ direct access to info as a great enhancement to teaching.
    • Only 4 of the 13 said they modified their classrooms in major ways.  Organized different, lectured less, relied on text less, gave students more independence, and acted more like a coach.  They became more student-centered in their teaching and made changes to their pedagogy.
    • Based on shadowing, most teachers used the basic teacher-centered instruction practice – lecturing, conducting a discussion, reviewing homework, working on assignments, and occassionally using overhead projectors and videos.  Even in computer based classes, teacher-centered instruction was the norm.
• Findings
  • General lack of usage among teachers in classrooms, labs, and media centers.  With access so high, the teachers in this study did not do much better than the national average of 2 of 10 teachers as serious users of technology, 4 of 10 as occassional users at least once per month.  This is bad because back in the 80’s, when all this tech wasn’t available, the national average for teachers as serious users was 1 in 10. 
  • Few fundamental changes in the dominant mode of teacher-centered instruction occurred.
  • Most teachers adapted technology to fit familiar practices of teacher-centered instruction.
  • Don’t know whether the classroom changes that occurred were due to technologies or whether they emerged as part of a gradual shift in their beliefs about teaching and tech provided a vehicle.
  • Reasons for low use:
    • Teachers don’t have the time to find and evaluate software.
    • Computer and software training was seldom offerred at convenient times.
    • Generic training available was irrelevant to teachers’ specific needs.
  • Teachers’ age, experience, and gender were not factors as there was little difference between veteran and novice teachers with or without previous tech experience, male or female.
  • Two ideas to explain the anomalies (mere speculation):
    • “Slow revolution” explanation – individuals and companies need decades to learn how to use and manage the new technology.  Increasing numbers of teachers will embrace integration of technology over time.
    • Historical context of high schools and their structures. (Much more likely.)
      • Embedded patterns of departments, 50 minute periods, diverse curricula, and other established practices are taken for granted and never really questioned by policy makers, practioners, researchers, etc.
      • Individual time constraints of teachers makes it difficult to find the time to integrate technology.
      • Teachers primarly responsibilities of covering material (and in the case today, high stakes testing) and other teaching demands.
      • Teachers good at integrating tech leave for or teaching positions or better jobs.
      • Defects of technology!  It needs to be reliable and it just isn’t there, causing a lack of confidence in technology.
      • Fundamental changes are required in how schools are organized, how time is allocated, and how teachers are prepared.
• Take-away
  • I wish this study was much bigger, but it does give some good insight into the gap between high access and low use of technology in school systems.
  •  TPCK anyone???????

Ausband, L.T. (2006). Instructional technology specialists and curriculum work. JRTE, 39(1), 1-21.

Abstract:

This case study investigated the job responsibilities of district-level instructional technology specialists that related to curriculum work and the perceptions the specialists had concerning their job responsibilities and their relationship to curriculum work. Data were collected through document analysis, shadowing, interviews, and a focus group. A framework of curriculum themes and categories was created, which was then used to define instructional technology work. Instructional technology specialists were found to be engaged in many aspects of curriculum work. The individual and focus group interviews revealed factors the participants considered to be barriers to getting their work done. Recommendations are provided for overcoming these barriers and a call is made to reconceptualize instructional technology specialists as curriculum workers.

o        RQs

1.      What professional duties do instructional technology specialists preceive they have that relate to curriculum work?

2.      What specific job responsibilities do instructional technology specialists have that relate to curriculum work?

3.      What are the barriers for instructional technology specialists and what possible changes can be made to overcome these barriers? 

This study is a very small study of 4 instructional technology specialists in a public school district in South Carolina.  The methods used in this study include: document collection (organization charts, job descriptions, syllabi, schedules, etc.), shadowing, individual interviews, focus group interviews, of which ONLY involved the instructional technology specialists.  The methods were used to provide information from different perspectives and allow for the triangulation of data.  Basically, the author examines the job responsibilities of the instructional technology specialists and explored their relationships with those of the curriculum workers.

The author of this study goes at length to compare the jobs of the instructional technology specialists to those of curriculum workers grounded in the work of Hamm, Pajak, and Sharp.  He determines that many of the responsibilites are the same, or at least very similar.  Based on research and instructional technology specialists perceptions, the author came up withe general responsibilites including:

• Developing curriculum and curriculum materials
• Collaborating with curriculum comittees and integrating technology into the curriculum
• Dealing with resources
• Visioning, strategic planning, and goal setting
• Collaborating with teachers, curriculum coordinators, and curriculm committees to integrate technology into instructional programs and meeting instructional needs
• Solving problems and researching and dispersing information
• Providing staff development and conducting teacher training
• Evaluating programs
• Instructional technology work
• Coordinating pedagogy and technology

Specifically, instructional technology specialists work with teachers to integrate technology into the curriculum through workshops and teaching courses, help teachers develop lesson plans that utilize technology, support teachers developing their portfolios, and attend meetings of Joint Department Leadership teams, content teams, School priority Teams, etc. acting in an instructional support role, along with other work.

The instructional technology specialists had many thoughts on barriers to their jobs.  They claimed to spend too much time on professional development and program evaluation activities than research, which they need to do to provide the right kinds of PD.  They noted frustration from inclusion in the actual decision-making processes, lack of time in schools, lack of communication, relationship and leadership issues between departments, accountability issues for integrating technology into schools, and a disconnect between technology and curriculum by teachers and district-level instructional support personnel.  “Teachers seem to think that if they have computer skills and are using those skills, they are integrating technology into their curriculm.”  “…Most of the curriculum specialists don’t have a clue about technology and don’t see it as part of the curriculum…”  They also believed that through helping teachers integrate technology into their curriculum that they also taught them how to use higher order thinking skills and  activities, manipulatives, etc.  It seems the major barrier for instructional technology specialists is in the organization of the schoool district.   Leadership, accountability, and communication issues mostly stem from different hierarchical management levels between constituents.  Curriculum workers have a different supervisor than instructional technology specialists whose supervisor is lower on the org chart than the curriculum workers.  When it comes time for getting goals and priorities, IT gets pushed to the back burner and/or faces more red tape.  Overall, the set up deams IT as much less important and are not included in many of the decision-making processes.

The author of this study makes some important points about technology and curriculum work.  He notes that “Technology integration is organizing the goals of curriculum and technology into a coordinated, harmonious whole.  True integration comes when students learn through computers, not about them” (Dockstader, 1999, p. 73).  Thus, we must understand how technology relates to curricular goals and how it supports the curriculum (needs study) and further research how technology work and curriculum work intersect.  At the very least, to be successful, these people need to work together on the same goals in close proximity to each other.  They may even need to background in both instructional technology and curriculum.  The author make the following recommendations:

• Instructional technology specialists need to be included in curriculum making and curriculum implementation decisions
• Instructional technology specialists should be located in the same building as curriculum workers.
• Curriculum workers should become more knowledgable about integrating technology into the curriculum, and about software and online resources that support the curriculum.
• Perhaps combine the two functions, insisiting that the employee be trained in both.
• Integrate documents used by teachers for planning, meaning combine both the technology and the curriculum planning documents that teachers use for their lessons.

Personal parting notes…  This is an interesting study because of my interest in a technology counselor.  Unfortunately, this study is so small, and the range in which districts are actually set up is so broad, but you have to start somewhere.  I agree with the intertwining of the functions of curriculum workers and instructional technology specialists.  If the IT person were to work hand in hand with the curricullum workers, it could potentially create a much better guide for teachers to integrate technology into the classroom without having to start from scratch.  It could also help develop a more uniform approach that could be tweaked based on the specific classroom environments and needs.  Additional help could be provided to those who need it and/or want it for other reasons (creative projects, etc.).  I wonder if the curriculum workers are actually teachers as well, as this could make it more difficult, especially considering time demands.  The down side of integrating IT and the curriculum worker as one position is the nature of specific educational subjects as well as great time demands.  You would have to have a whole additional set of people for day to day support (paraprofessionals?)

Hickey, D. T., Moore, A. L., & Pellegrino, J. W. (2001). The motivational and academic consequences of elementary mathematics environments: Do constructivist innovations and reforms make a difference? American Educational Research Journal, 38(3), 611-652. CARET summary.

Abstract: This study examined the effects of a videodisc-based mathematical problem- solving series known as The Adventures of Jasper Woodbury, as imple- mented by one school district within a constructivist-inspired reform of its math curricula. The motivational and academic consequences of both the specific innovation and the broader reforms were examined in 19 fifth- grade classrooms in two pairs of closely matched schools. Onepair of schools served higher-achieving high-socioeconomic status (SES) students while the other pair served relatively lower-achieving low-SES students. Significantly larger gains on the Mathematical Problem-solving subtest of the ITBS were documented in the 10 classrooms where the Jasper activities were implemented, and in the 10 classrooms that were ranked as relatively more con- sistent with the broader curricular reform goals. The largest relative gains were found in the five classrooms that both used the Jasper activities and were ranked more consistent with the broader reforms. The positive consequences of both the Jasper activities and the broader reforms were documented in both pairs of schools. The implications of these results are discussed relative to currentproposals for curricular reform and research on educational innovations.

Source:
 
National Research Council of the National Academies.  (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press.  (Chapters Introduction, 5-8).

Chapter 8 - Teaching and Learning Functions (Mindy Kalchman and Kenneth R. Koedinger)

1.      Addressing the 3 Principles

1.1.   Currently, teaching instruction is very procedural based and does not take into account the 3 principles of HSL, especially not taking into account the importance of conceptual understanding. (p. 353)

1.2.   Teaching, including instructional approaches and curricula, needs to reflect the HSL principles in order for students to be able to learn and understand the complex mathematical concept of functions. (p. 358)

1.3.   Principle #1: Building on Prior Knowledge

1.3.1.      Emphasizes the importance of students and teachers continually making links between students’ experiences outside the math classroom and their school learning experiences

1.3.2.      Understandings students bring can be viewed in 2 ways and teaching should make use of both:

1.3.2.1.Everyday, informal, experiential, out-of-school knowledge

1.3.2.2.School-based or “instructional” knowledge

1.3.3.      Students in this instructional approach are introduced to function in the context of a walkathon chosen because:

1.3.3.1.Students are familiar with money and distance as variable quantities.

1.3.3.2.They understand the contingency relationship between variables.

1.3.3.3.They are interested in and motivated by the rate at which money is earned.

1.3.4.      It is critical to use a powerful instructional context, called a “bridging context”, which serves to bridge students’ numeric(equations) and spatial(graphic) understandings and link their everyday experiences to lessons in the mathematics classroom.

1.3.5.      Walkathon context as introduction to functions in multiple forms.

1.3.5.1.Walkathon = real world

1.3.5.2.Table, graph for representations.

1.3.5.3.$1 for each kilometer equiv to $1 = 1*km equiv to y = x*1 (p. 359)

1.3.6.      Slope

1.3.6.1.The topic is usually reserved for 9^th grade mathematics and is part of students’ introduction to relations and functions in general and to linear functions in particular.

1.3.6.2.Defined as ratio of vertical distance to horizontal difference, or “rise to run”.

1.3.6.3.Slope = rise/run

1.3.6.4.Once the equation for a line has been introduced (y=mx+b), m is defined as the slope and is calculated using m = (y2 – y1)/(x2 – x1)

1.3.6.5.For students to understand slope in these definitional and symbolic ways, they must have a lot of formal knowledge, including: (p. 362)

1.3.6.5.1.      Meaning of ratio, coordinate graphing, variables, subscripts

1.3.6.5.2.      Skills such as solving equations in 2 variables and combining arithmetic operations.

1.3.6.6.Knowing the definition does not ensure actual understanding of the meaning of slope with regard to steepness or rate of change.

1.3.6.7.Younger students have intuitive and experiential understandings of slope that can be used as a basis for formal learning that involves conventional notations, algorithms, and definitions.

1.3.6.7.1.      When given a word problem, students had a better chance of finding the slope without receiving instruction than students given the textbook definition of slope.

1.3.6.7.2.      This is not to say that problems should be phrased in “student language”, as it is important for students to learn formal mathematic terminology and abstract algebraic symbolism.

1.3.6.7.3.      The point is that using student language is a way to assess what knowledge students are bringing to a topic, and then linking and building on what they know to guide them towards a deeper understanding of formal mathematical terms, algorithms, and symbols.

1.3.7.      Students’ prior knowledge acts as a building block for development of more sophisticated ways of thinking mathematically.

1.3.8.      We may sometimes underestimate the knowledge and skills students bring to the learning of functions (p. 363) that may serve as basis for formalizing students’ thinking.

1.4.   Principle #2: Building Conceptual Understanding, Procedural Fluency, and Connected Knowledge

1.4.1.      The purpose is to simultaneously develop conceptual understanding and procedural fluency, and helping students connect and organize knowledge in various forms.

1.4.2.      Students can develop surface facility with notations, words, and methods of domain study without having a foundation of understanding.  This prevents students from utilizing mathematics and making number sense in the real-world.

1.4.3.      Want students to understand the core concept of a functional relationship: that the value of one variable is dependent on the value of another.

1.4.4.      Want students to understand that a function can be expressed in a variety of ways, which all have the same meaning (graphs, equations, tables, etc.)

1.4.5.      Need to deliberately build and secure that knowledge so that a student can confidently tackle sophisticated problems.

1.4.6.      Good teaching requires a solid understanding of the content domain and specific knowledge of student development.

1.4.7.      Developmental model that this instructional approach is based on has 4 levels (0-3) each describing what students can typically do at a given developmental stage.

1.4.7.1.Level 0

1.4.7.1.1.      Characterizes the kinds of numeric/symbolic and spatial understandings students typically bring to learning functions.

1.4.7.1.2.      Initially numeric and spatial understandings are separate.

1.4.7.1.3.      Initial numeric understanding – students can iteratively compute within a string of positive whole numbers. (p. 364)

1.4.7.1.4.      Initial spatial understanding – students can represent relative sizes of quantities on graph and perceive patterns of qualitative changes in amount by left-to-right visual scan of the graph, but cannot quantify those changes.

1.4.7.2.Level 1

1.4.7.2.1.      Students begin to elaborate and integrate their initial numeric and spatial understandings of functions.

1.4.7.2.2.      Numeric understanding:

1.4.7.2.2.1.Students iteratively apply a single operation to, rather than within, a string of numbers to generate a second string of numbers.

1.4.7.2.2.2.Construct an algebraic expression for this repeated operation.

1.4.7.2.3.      Spatial Understanding

1.4.7.2.3.1.Students progress from understanding graphs with verbal or categorical values along the x-axis to understanding graphs with quantitative values along the x-axis. (p. 367)

1.4.7.2.3.2.Use continuous quantities along a horizontal axis.  Without this, students cannot see graphs as representing the relationship between 2 changing quantities. (p. 367)

1.4.7.2.3.3.Perceive emergent properties, such as linear or increasing, in the shape of the line drawn between points.

1.4.7.2.4.      Integration of elaborated understandings (p. 365) – See the relationship between the differences in the y-column in a table and the size of the step from one point to the next in the associated graph.

1.4.7.2.5.      Interpret algebraic representations both numerically and spatially.

1.4.7.2.6.      Grasping the why and how the line on a graph maps onto the relationship described in a word problem or equation is a core conceptual understanding.  If students’ understanding is only procedural, they will not be well prepared for the next level. (p. 368)

1.4.7.3.Level 2

1.4.7.3.1.      Elaborate initial integrated numeric and spatial understandings to create more sophisticated variations.

1.4.7.3.2.      Integrate understanding of y=x and y=mx + b to form a mental structure for linear functions.

1.4.7.3.3.      Integrate rational numbers and negative integers.

1.4.7.3.4.      Form mental structures for other families of functions, such as y=x^n + b.

1.4.7.4.Level 3

1.4.7.4.1.      Integrate variant (e.g. linear and nonlinear) structures developed at level 2 to create higher-order structures for understanding more-complex functions, such as polynomials and exponential and reciprocal functions.

1.4.7.4.2.      Elaborate understandings of graphs and negative integers to differentiate the four quadrants of the Cartesian plane

1.4.7.4.3.      Understand the relationships of these quadrants to each other. (p. 366)

1.4.7.4.4.      Without the development of a sound conceptual frame work for functions, students do not make sense of their work.  For example, student’s graph has a negative slope and a negative y-intercept and his table had values without a constant rate of change for a function with a y-intercept of -10 and a positive slope.  Student did not recognize the impossibility.

1.4.7.4.5.      A teacher can help support integration of concepts by moving fluidly and rapidly between numeric and spatial representations.  This helps students simultaneously build understandings of each representation individually, and in an integrated nature.

1.4.7.4.6.      Integration helps students begin to understand and organize their knowledge in ways that facilitate the retrieval and application of relevant mathematical concepts and procedures.

1.4.7.4.7.      If students’ spatial and numeric understandings are not integrated they may not notice when conclusions drawn from each understanding are inconsistent. (p. 369)

1.5.   Principle #3: Building Resourceful, Self-Regulating Problem Solvers

1.5.1.      Instruction should assist students not only with mathematical procedures and concepts, but also in thinking about procedure and concepts in reflecting on and articulating their own thinking and learning.

1.5.2.      Encouraging students to reflect on and communicate their ideas about functions supports (p. 371) them in making connections among representations that are necessary for flexible, fluent, and reliable performance.

1.5.3.      An especially important type of metacognitive thinking in mathematics is coordinating conclusions drawn from alternate mathematical representations.

1.5.3.1.Solve problem in more than one way.

1.5.3.2.Checking consistency of verbal interpretations of different representations. (Ex: Increasing versus decreasing slope.)

1.5.3.2.1.      Encourages students to think about problems in (p. 372) multiple ways (strategies) and with multiple tools (representation).

1.5.3.2.2.      Encourages students to draw conclusions that are not only quantitative, but also qualitative (verbal interpretations).

1.5.4.      Must also create an atmosphere where students feel comfortable to explore, experiment, and take risks in problem solving and learning.

1.5.5.      Teacher must also help students develop a tolerance for the difficulties mathematics sometimes presents.

1.5.6.      Teacher must also help students learn when they’ve reached their limits in understanding and how to ask for help or find help on their own (peer, computer, calculator, etc.)

2.      Teaching Functions for Understanding

2.1.   In addition to knowledge of the content being taught and a development model of how students acquire understanding of the content, an instructional set of strategies for moving students along the developmental pathway and addressing obstacles and opportunities that appear most frequently on the way is required.

2.2.   The below method has been successful in increasing understanding of functions for 8^th and 10^th graders.  In fact 6^th graders taught with this approach were more successful on a function test than 8^th and 10^th graders taught with conventional instruction.

2.3.   Curriculum For Moving Students Through the Model

2.3.1.      Encourages students to:

2.3.1.1.Build upon and apply prior knowledge. (p. 373)

2.3.1.2.Construct an integrated conceptual framework for understanding functions.

2.3.1.3.Apply metacognitive skills to their learning.

2.3.2.      Used effectively for students in 6^th, 8^th, 10^th, 11^th grades.

2.3.3.      Unit requires approximately 650 minutes of class time to complete.

2.3.4.      Regardless of grade, it is recommended that you go through the whole sequence of lessons in order to develop and utilize deeper and proper understandings.

2.3.5.      This instruction is different than traditional instruction in that it uses contextual bridges as opposed to different contexts for different situations.  This allows for an easier understanding of relationships between representations.

2.3.6.      This instruction also differs in that it focuses on multiple representations (co-presented) instead of singular ones.  In traditional instruction, singular representations may be taught for weeks before different representations are related.

2.3.7.      Engages students in the construction of functional notations and helps them build notations and meaning for constructs.  Traditional instruction just gives students formal notation (p. 374) and then gives them procedures for finding the “answers”.

2.3.8.      Over the course of instruction, students progressively formalize their initial notations until those notations correspond with conventional general equations (such as y = mx + b).

2.3.9.      Suggests follow-up activities that allow students to remain situated in the context of instruction for the first part of the unit until they are confident and competent with the concepts on a more abstract basis.

2.3.10.  Then when students move up to the computer environment, no new concepts are introduced initially.  Students first have time to consolidate the individual concepts of the first part of the unit, then move on to more challenging activities that advance their thinking and understanding in the domain.

2.3.11.  Students give presentations on a particular kind of function to their classmates in order to show their understanding and share their expertise in key characteristics and behaviors of those functions. (p. 375)

2.3.12.  Example Lessons

2.3.12.1.        Level 1

2.3.12.1.1.  Introduction – Walkathon

2.3.12.1.1.1.  Description – Students record in tables the money earned for each km walked and plot each pair of values for a variety of rules.  An equation is formed using km and $ based on the rule of sponsorship.

2.3.12.1.1.2.  Activities – Student pairs invent 2 of their own sponsorship arrangements for which their partner constructs tables, graphs, and equations.

2.3.12.1.2.  Slope

2.3.12.1.2.1.  Description – Introduced as the up-by (or down-by) amount between successive $ values in a table or graph.  It is the relative steepness of a function and the amount by which each km is multiplied.

2.3.12.1.2.2.  Lesson – big focus on prior knowledge(p. 379)

2.3.12.1.2.2.1.  Approx 90 minutes (p. 379)

2.3.12.1.2.2.2.  Suggest beginning with the graph and the table for the rule of earning $1 for every km walked. (p. 379)

2.3.12.1.2.2.3.  Draw the students’ attention to the fact that the up-by amount corresponds to the mathematical concept of slope and that is a relative measure of a function’s steepness. (p. 379)

2.3.12.1.2.2.4.  Use y=x as a conceptual landmark. (p. 379)

2.3.12.1.2.2.5.  Challenge students to provide sponsor rules that are steeper, less steep than y=x.  Encourage students to plot them on the same set of axes.  Ask students to predict the steepness. (p. 379)

2.3.12.1.2.2.6.  Ask students to invent other rules and make tables and graphs for them. (p. 380)

2.3.12.1.2.2.7.  Ask students to summarize their findings. (p. 380)

2.3.12.1.2.2.8.  Ask students how the steepness of a function can be quantified. (p. 380)

2.3.12.1.2.2.9.  Introduce working with negative slopes by using the perspective of the person donating the $ (sponsor). (p. 380)

2.3.12.1.2.3.  Activities – Students are asked to find the slope of several different functions expressed in tables, graphs, and equations.

2.3.12.1.3.  Y-Intercept

2.3.12.1.3.1.  Description – Introduced as the “starter-offer”, a fixed starting bonus students receive before the walkathon begins.  It affects only the vertical starting point of the numeric sequence and graph.  It does not affect the steepness or shape of the line.

2.3.12.1.3.2.  Lesson (p. 381)

2.3.12.1.3.2.1.  Theory-based instructional design in connecting students’ factual/procedural and conceptual knowledge. (p. 381)

2.3.12.1.3.2.2.  Instead of using x=0, it focuses on other numbers to develop a better understanding. (p. 381)

2.3.12.1.3.2.3.  Approx 90 minutes (p. 381)

2.3.12.1.3.2.4.  Bonus or additional amount of money sponsored before the walkathon begins – “starter-offer”. (p. 381)

2.3.12.1.3.2.5.  Students are asked to construct tables of values and then graph them, then formulate an equation.  (Ex. $5 bonus, and $1 for every km) (p. 381)

2.3.12.1.3.2.6.  Have them verbally describe the relationship to help formulate the equation. (p. 381)

2.3.12.1.3.2.7.  Offer new starter amounts with $1/km and ask students to predict where each new function will be on the graph, then construct the new tables, graphs, and equations. (p. 382)

2.3.12.1.3.2.8.  Students are asked to describe patterns and characteristics they see. (p. 382)

2.3.12.1.3.2.9.  Want students to see that the only effect of changing the starting offer is a vertical shift in the graph.  Students tend to confuse slope and y-intercept.  (p. 382)

2.3.12.1.3.2.10.    Negative y-intercepts are introduced using the concept of debt. (p. 382)

2.3.12.1.3.2.11.    Allow students to be informal in their notation until they are able to consolidate and formalize the concepts. (p. 383)

2.3.12.1.3.2.12.    In lessons on non-linear functions, the starter-offer idea is also applied. (p. 383)

2.3.12.1.3.3.  Activities  - Students invent 2 linear functions that allow them to earn exactly $153 after walking 10 km.  Students record the slope and y-intercept of each function and explain how the y-intercept of each function can be found in its table, graph, and equation.

2.3.12.1.4.  Curving Functions

2.3.12.1.4.1.  Description – Nonlinear functions are introduced as those having up-by amounts that increase (or decrease) after each km walked.  They are derived by multiplying the km (x) by itself at least once.  The more times x is multiplied by itself, the greater is the difference between $ values and thus the steeper the curve.

2.3.12.1.4.2.  Activities – Students are asked to decide which of 4 functions expressed in tables are nonlinear and explain their reasoning.  Asked to write an equation for and to sketch the label the graph of each function.  Students were asked to come up with a curved-line function for earning $153 over 10 km. (p. 376)

2.3.12.2.  Level 2 and 3

2.3.12.2.1.  Computer Activities

2.3.12.2.1.1.  Description – Use spreadsheet technology and prepared files and activity sheets to consolidate and extend the understandings about slope, y-intercept, and linearity (level 2). Work in all 4 quadrants to transform quadratic and cubic functions and explore their properties, behaviors, and characteristics of exponential, reciprocal, and other polynomial functions.

2.3.12.2.1.2.  Lesson – big on metacognitive abilities (p. 387)

2.3.12.2.1.3.  Activities – Students change the steepness, y-intercept, and direction of y=x and y=x^2 to make the function go through preplotted points.  They record the numeric, algebraic, and graphic effects of their changes.  They invent functions with specific attributes, such as parallel and y-intercept below x-axis, or inverted parabola that is compressed and in the lower left-hand quadrant.

2.3.12.2.2.  Presentations

2.3.12.2.2.1.  Description – groups of students investigate and then prepare a presentation about a particular type of function.  This stimulates discussion and summarization of key concepts and serve as a partial teacher assessment for evaluating students’ post-instruction understanding about functions.

2.3.12.2.2.2.  Activities – Groups of students use computer generated output of graphs, equations, and tables to illustrate a particular type of function’s general properties and behaviors.  Students give presentations about their function and share expertise with their classmates. (p. 377)

2.4.   Alternate development models

2.4.1.      Jasper Woodbury Series – more complex, real-world contexts

2.4.2.      Cognitive Tutor Algebra (previously called PUMP) – uses multiple real-world contexts and computer support.

Source:
 
National Research Council of the National Academies.  (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press.  (Chapters Introduction, 5-8).

Chapter 7 - Pipes, Tubes, and Breakers: New Approaches to Teaching the Rational Number System (Joan Moss)

1.      Difficulties of the Rational Number System

1.1.   Extensive research shows that many people, adults and children alike, having difficulty with the rational number system (fractions, decimals, etc.).

1.2.   Introduced in early elementary school, students must reformulate the way they think about numbers in a major way.  They must go beyond whole-numbers ideas, which are expressed in fixed quantities, to understand numbers expressed in relationship to other numbers.

1.3.   New relationships are grounded in multiplicative reasoning very different from additive reasoning that characterizes whole numbers.

1.4.   The majority doesn’t make the transition smoothly, and become disenchanted and frustrated with mathematics.

1.5.   A common misunderstanding is continued use of whole-number reasoning in situations where it does not apply. Examples:

1.5.1.      Children say .059 > .2

1.5.2.      Children say 1/3 + ½  = 1/5 (even many 5^th and 6^th graders)

1.5.3.      Children say 1/8 > 1/6

1.6.   Rational number system introduces a major stumbling block in children’s mathematical development.

1.7.   It marks a time where students no longer know what is going on in the math classroom, and it is disheartening to both students and teachers.

1.8.   It is a cause for concern because rational numbers are an underlying basis of advanced mathematics.

1.8.1.      Students cannot succeed in advanced mathematics (ex. algebra) without rational numbers.

1.8.2.      The rational number system is also a major part of our daily adult lives. (p. 310)

2.      Rational-Number Learning and the Principles of How People Learn

2.1.   The Knowledge Network: New Concepts of Numbers and New Applications (Principle 2) (p. 312)

2.1.1.      Rational numbers is a new knowledge network – it’s filled with new and intertwined concepts, new facts, and new symbols.

2.1.2.      Based on multiplicative instead of whole-number relations.

2.1.3.      Operations involved may appear less intuitive and at odds with earlier understandings.

2.1.4.      New Symbols, New Meanings, New Representations

2.1.4.1.A specific rational number can take several many forms (i.e. fraction, decimal, percent)

2.1.4.2.A rational-number quantity can be represented by an infinite number of equivalent common and decimal fractions. (1/4, 2/8, 4/16, ….25, .250,

2.1.4.3.For many students, flow between representations does not come easily (p.313).

2.1.4.4.Most traditional instruction of decimals, fractions, and percents is taught separately as distinct topics, which makes it confusing and more difficult for students to realize that a single quantity can have many representations.

2.1.5.      Subconstructs

2.1.5.1.A single rational number can have several conceptually distinct meanings.

2.1.5.2.Example = ¾

2.1.5.2.1.      Part-whole relationship, as in 3 of 4 equal size shares.

2.1.5.2.2.      Quotient interpretation, as in 4 kids sharing 3 pies. (p. 314)

2.1.5.2.3.      Ratio, as in 3 red cars for every 4 green cars, not to be confused by the part-whole relationship of 3/7, where 3 out of the 7 cars are red.

2.1.5.2.4.      Measure, as in ¾ of an inch.

2.1.5.2.5.      Multiplicative operator, behaving as an operation that reduces or enlarges the size of another quantity, as in ¾ of the original size.

2.1.5.3.Coordinating these different interpretations requires a deep understanding of concepts and interrelations among them.

2.1.6.      Reconceptualizing the Unit Operations

2.1.6.1.Rational numbers are dense – we can find an infinity of other numbers between any 2 rational numbers.

2.1.6.2.Rational numbers are also implied, as opposed to explicit. (p. 315)

2.1.7.      New Conceptualizations

2.1.7.1.Numbers must be understood in multiplicative relationship.

2.1.7.2.Students must understand that numbers are no longer independent, and understand a fraction as a new kind of quantity that is defined multiplicatively by relative amount conveyed by symbols.

2.1.7.3.Looking at students prior conceptions and relevant understandings can help support the conceptual change.

2.2.   Students Errors and Misconceptions Based on Previous Learning (Principle 1)

2.2.1.      Conceptions of numbers are grounded in whole-number learning, which leads them to confusion in rational numbers. (p. 316)

2.2.2.      For example, many students think that 2/3 and ¾ are equal because “one piece” is missing.  They are thinking in additive, absolute terms instead of relative or proportional (multiplicative) terms. (p. 317)

2.3.   Metacognition and the Rational Number (Principle 3)

2.3.1.      Students must be actively engaged in sense making to solve problems competently.

2.3.2.      Most middle school children do create appropriate meanings for fractions, decimals, and percents; instead they rely on memorized rules for symbol manipulation.

2.3.3.      Mistakes are made by students’ failure to monitor their operations and judge the reasonableness of their responses.

2.3.4.      Classroom teaching must support students in developing metacognitive skills (explaining their answers, strategies, interpretations, etc.) or math may no longer make sense to students, particularly at the start of rational numbers.

3.      Instruction in Rational Number

3.1.   Why does instruction so often fail to change students’ whole-number perceptions?

3.1.1.      HSL principles are routinely violated.

3.1.2.      Topics in rational numbers are typically covered quickly and superficially, but it takes time for students to master it thoroughly.

3.1.3.      Not enough time is dedicated to teaching conceptual meaning behind rational numbers, instead procedures receive greater emphasis.

3.1.4.      Procedural competence is important, but it must be anchored by conceptual understanding, and this is not the case for a great number of students.

3.1.5.      Other aspects, such as teaching the notation system for decimals, are short-changed in this way as well.

3.1.6.      Operations tend to be taught in isolation, (p. 319) divorced from meaning.

3.1.7.      Almost no time is spent relating various representations – fractions, decimals, percents – to each other.

3.1.8.      Researchers agree that textbooks fail to provide a grounding for the major conceptual shift to multiplicative reasoning that is essential to mastering rational number.

3.1.9.      Examples of instructional approaches

3.1.9.1.Pie Charts and a Part-Whole Interpretation of Rational Numbers

3.1.9.1.1.      Typically a pie or cake is used to introduce fractions.  The object is divided into equal pieces, with a certain number of pieces shaded.  Students count the number of pieces which indicates the denominator, then count the shaded pieces, which becomes the numerator. (p. 320)

3.1.9.1.2.      This instruction is an obvious approach, but it is an additive approach and does not introduce children to the difficult conceptual shift of multiplicative reasoning.

3.1.9.2.Alternative Instructional Approaches: Ratio and Sharing

3.1.9.2.1.      Multiplicative operations of splitting.

3.1.9.2.1.1.Folding paper rather than pie charts. (Kieren)

3.1.9.2.1.2.Contexts of ratios using cooking, shadows, gears, and ramps. (Confrey)

3.1.9.2.1.3.Contexts of ratios using equal shares and quotients by having children use realistic situations like sharing pancakes or chocolate bars and the children devise a system themselves.  (Streefland) (p. 321)

3.2.   Cirriculum developed using a different approach to rational number, alongside Robbie Case, shown through controlled experiment trials to be effective in helping students in the 4^th, 5^th, and 6^th grades gain a strong initial grounding in the number system, highlights multiplicative understanding, with an additional focus on interrelations among fractions, decimals, and percents.

3.2.1.      Pipes, Tubes, and Breakers: A New Approach to Rational Number Learning

3.2.1.1.Percents as a stating point

3.2.1.1.1.      Start with percents first because it only involves fractions of the base 100.

3.2.1.1.2.      Done through everyday understanding of students.

3.2.1.1.3.      Students are challenged to consider relative lengths of different quantities.

3.2.1.1.4.      Initial activities direct students’ attention to ideas of relative amount and proportion from the very beginning of their learning of rational number. (p. 322)

3.2.1.1.5.      Ex: 2 different size beakers both 50% full.

3.2.1.1.6.      Ideas of percents and proportion serve as the anchoring concept for subsequent learning of decimals and fractions, then for understanding of the number system as a whole.

3.2.1.2.Starting Point: Visial Proportional Estimation and Halving and Doubling

3.2.1.2.1.      From prior experiences and understandings, students at this age have generally developed the ability to estimate proportions, such as halves, visually, and have the ability to work with successive halving. (p.323)

3.2.1.2.2.      Both abilities have grounding in multiplicative operations.

3.2.1.2.3.      Idea is to merge these separate understandings to construct conceptual grounding for rational numbers.

3.2.1.2.4.      Strategy was to develop a “bridging context” to help students first access their knowledge of visual proportions, then integrate it with their knowledge of halving.

3.2.1.2.5.      Context is having students work with percents and linear measurement, allowing students to access initial kind of understanding and integrate them in a natural fashion.

3.2.1.3.Why Percent as a Starting Point?

3.2.1.3.1.      Students always working with the denominator of 100, thus postponing problems that arise when students must compare or manipulate ratios with different denominators.

3.2.1.3.2.      Allows students to concentrate on working with their own procedures rather than struggling to master complex algorithms or procedures.

3.2.1.3.3.      All percentages have a corresponding decimal or fractional equivalent that can be relatively easy to determine, which helping in starting to develop the understanding of how the 3 representations are related.

3.2.1.3.4.      Children know a good deal already about percents from their everyday experiences. (p. 324)

3.2.2.      Curriculum Overview

3.2.2.1.Divided into roughly 3 parts.

3.2.2.1.1.      Percent – uses concrete props that highlight linear measurement.

3.2.2.1.2.      Two-place decimal – introduced like the percent of the way between 2 whole numbers.

3.2.2.1.3.      Activities that promote comparing and ordering rational numbers and moving along decimals and percents.  Fractions are also taught at this stage in relation to percents and decimals.

3.2.2.2.Lessons

3.2.2.2.1.      Lessons Part 1: Introduction to Percents

3.2.2.2.1.1.Percents in Everyday Life – discussion to elicit what students know about the topic.

3.2.2.2.1.2.Pipes and Tubes: A Representation of Fullness

3.2.2.2.1.2.1.      Props include black drainage pipes and white venting tubes that can be raised or lowered simulating (p. 325) the action of water filling them to different levels.

3.2.2.2.1.2.2.      To get a better idea of the students’ understanding, they are asked how they would teach percent to another child using these props.

3.2.2.2.1.2.3.      Children are asked to use the props to generate ideas about percents above 100.

3.2.2.2.1.3.Percents on Number Lines: More Estimations

3.2.2.2.1.3.1.      Included activities with laminated meter-long number lines calibrated in centimeters to provide students with another way of visualizing percent. (p. 327)

3.2.2.2.1.3.2.      Activities included using the line as a sidewalk where students were asked by peers to walk some %.

3.2.2.2.1.3.3.      Activities are used to consolidate percent understanding and extend linear measurement context.

3.2.2.2.1.4.Computing with Percent

3.2.2.2.1.4.1.      Estimate the fullness of various beakers of water.

3.2.2.2.1.4.2.      Leads naturally to a focus on computation and measurement using halving strategies.

3.2.2.2.1.5.Invented Procedures – From the previous activity, students used their halving strategy as the basis for their procedure of calculating certain percents (e.g. 75%).

3.2.2.2.1.6.String Challenges: Guessing Mystery Objects

3.2.2.2.1.6.1.      String measurement activities as a way of considering percent quantities and calculating percentages using benchmarks.

3.2.2.2.1.6.2.      “The Mystery Object Challenge” – teacher held up a piece of string that was cut to the percent of the length of a certain object in the room. (p. 329)  Eventually, the students were invited to pick the object to challenge the rest of the class.

3.2.2.2.1.6.3.      Students then created “percent family” posters of strings using their mystery object as the base, which provided another opportunity for calculating percents, as well as reinforced the idea of proportion.  “Our string lengths are different even though all our percents are the same.”

3.2.2.2.1.7.Summary of Lessons Part 1

3.2.2.2.1.7.1.      Estimations then calculations of percent quantities.

3.2.2.2.1.7.2.      Presented in the context of linear measurement of pipes, tubes, beakers, string, and number lines.

3.2.2.2.1.7.3.      Students were not given formal instruction, but naturally employed procedures of their own.

3.2.2.2.1.7.4.      While percent was the only form of rational numbers introduced, students often referred to fractions when working these initial activities. (p. 331)

3.2.2.2.1.7.5.      Students were told that 12 ½ % is 1/8, as they new 25% is ¼.

3.2.2.2.1.7.6.      Activities continually helped students integrate their sense of visual proportion with their ability to do repeated halving and help build the foundation for further learning of this number system.

3.2.2.2.2.      Lessons Part 2: Introduction of Decimals

3.2.2.2.2.1.Two-place decimal.

3.2.2.2.2.2.Initial lessons also had strong focus on measurement and proportion.

3.2.2.2.2.3.Research has shown that a solid conceptual grounding in decimal numbers is hard for a student to achieve.

3.2.2.2.2.4.Makes a direct link from percents to decimals.

3.2.2.2.2.5.Show students that a two-place decimal number represents a percentage of the way between 2 adjacent whole numbers, or an intermediate distance between 2 numbers.

3.2.2.2.2.6.Decimals and Stopwatches

3.2.2.2.2.6.1.      Props for students were LCD stopwatches displaying seconds and hundredths of seconds.

3.2.2.2.2.6.2.      Students were asked to consider what the 2 “smaller numbers” might mean.

3.2.2.2.2.6.2.1.Students noted that there were 100 of these time units.

3.2.2.2.2.6.2.2.“It’s like they are percents of a second.” (p. 332)

3.2.2.2.2.6.3.      Asked to name the time units and it was termed as the quantity they understood to mean time that had passed between any 2 whole seconds.

3.2.2.2.2.6.4.      Work continued with the stopwatches and decimals with a focus on ordering numbers.

3.2.2.2.2.7.Magnitude and Order in Decimal Numbers

3.2.2.2.2.7.1.      To illuminate the difficult concepts of magnitude and order, activities were designed to help students with ordering decimals.

3.2.2.2.2.7.2.      “Stop-Start Challenge”

3.2.2.2.2.7.2.1.Students attempted to start and stop the stopwatch as quickly as possible, several times in succession.

3.2.2.2.2.7.2.2.After discussion, they learned to record their times as decimals.

3.2.2.2.2.7.2.3.Students then compared their reaction times with their peers and ordered their times.

3.2.2.2.2.7.2.4.Students can learn from their experience in trying to get the quickest time that .09 is small than .10.

3.2.2.2.2.7.3.      “Stop the Watch Between” is another game to engage students in the issue of magnitude. (p. 333)

3.2.2.2.2.7.4.      Laminated number lines where students were asked to indicate parts of 100 using decimal representations.

3.2.2.2.2.8.Summary of Lessons Part 2

3.2.2.2.2.8.1.      First understand how decimals and percents are related, then learn how to represent decimals symbolically.

3.2.2.2.2.8.2.      Activities proceeded for students to consider and reflect on magnitude.

3.2.2.2.2.8.3.      Finally students engaged in comparing and ordering decimals.

3.2.2.2.2.8.4.      First step in students’ learning to translate among the representations of rational number and gain fluency with different kinds of operations.

3.2.2.2.3.      Lessons Part 3: Fractions and Mixed Representations of Rational Numbers

3.2.2.2.3.1.Fractions First: Equivalencies

3.2.2.2.3.1.1.      Give students a chance to work with fractions more formally and then provide them with opportunities to translate flexibly among fractions, decimals, and percents.

3.2.2.2.3.1.2.      Students were asked to represent a fraction in as many ways as they could. (Ex: 3/4 they would represent as 6/8, 75/100).

3.2.2.2.3.1.3.      They were also asked to incorporate fractions into word problems they created and other students solved.

3.2.2.2.3.1.4.      Students also created equations with questions for other students to solve. (Ex: How many to make a whole for 1/8 + ½ + ¼?) (p. 334)

3.2.2.2.3.1.5.      Students initially used fractions in the equations, but then started adding percents and decimals on their own.

3.2.2.2.3.2.Crack the Code

3.2.2.2.3.2.1.      Further work with LCD stopwatches.

3.2.2.2.3.2.2.      Students moved between representations of rational numbers as they were challenged to stop the watch at the decimal equivalent of different fractions and percents.

3.2.2.2.3.2.3.      This allowed them to increase their understanding of the possibility of fluid movement between representations.

3.2.2.2.3.3.Card Games

3.2.2.2.3.3.1.      Specially designed set of cards depicting various representations of fractions, decimals, and percents.

3.2.2.2.3.3.2.      Students used the cards to design games tat challenged their peers to make comparisons among and between representations. (p. 335)

3.2.2.2.3.4.Summary of Lessons Part 3

3.2.2.2.3.4.1.      Focused primarily on students’ uses of mixed representations.

3.2.2.2.3.4.2.      Started with some formal activities with fractions and equivalencies, then had students make up own games and challenges for more practice in flexible movement from one operation to another.

3.2.2.2.3.4.3.      One of the primary goals was to provide students with habits of mind regarding multiple representations that will be with them throughout their learning and lay the foundation for their ability to solve mathematical problems.

3.2.2.3.Results from Studies

3.2.2.3.1.      Needed to look at the improvement made by individual students at the end of the experimental intervention.

3.2.2.3.2.      Also interested to see the performance of students in experimental group compared to those with traditional classroom instruction.

3.2.2.3.3.      Thus, assessed the experimental students on a variety of task before and after course instruction and administered these same tasks (p. 337) to students from classrooms with standard textbook instruction.

3.2.2.3.4.      Students in experimental group improved significantly, often higher than those students receiving traditional classroom instruction, who were often also older.

3.2.2.3.5.      Quality of answers was also better from the experimental group, and also made more frequent reference to proportional concepts. (p. 338)

3.2.2.3.6.      Knowledge Network

3.2.2.3.6.1.Students gained the following understandings toward developing their knowledge network:

3.2.2.3.6.1.1.      An overall understanding of the number system, as evidenced by their ability to use representations of decimals, fractions, and percents interchangeably.

3.2.2.3.6.1.2.      Appreciation of the magnitude of rational numbers, as seen in their ability to compare and order numbers.

3.2.2.3.6.1.3.      Understanding of the proportional- and ratio-based constructs of this domain, demonstrated by equivalencies.

3.2.2.3.6.1.4.      Understanding of percent as an operator, as demonstrated by their ability to invent a variety of solution strategies for calculating with these numbers.

3.2.2.3.6.1.5.      General confidence and fluency in their ability to think about the domain using the benchmark values they had learned, which is a hallmark of number sense.

3.2.2.3.6.2.Research is still in an early stage.

3.2.2.3.6.3. Research will continue to pursue questions, including the potential limitations of continued halving as a way of operating with rational numbers, downplaying the importance of quotient subconstruct, as well as a limited view with fractions.

3.2.2.3.6.4.Research will also continue to learn more about how children learning this way should proceed with their learning of mathematics. (p. 340)

Source:
 
National Research Council of the National Academies.  (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press.  (Chapters Introduction, 5-8).

Chapter 6 – Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades (Sharon Griffin)

1.      Teaching Mathematics in the Primary Grades (This chapter focuses on Number Sense)

1.1.   You need to know 3 things:

1.1.1.      Where you are now (in terms of students’ prior knowledge).

1.1.2.      Where you want to go (knowledge you want children to acquire).

1.1.3.      The best way to get there.

1.2.   Each leads to a question crucial to the design of effective mathematics instruction.

1.3.   Each points to a body of knowledge the teacher must have access to in order to answer the question.

1.4.   Similar to the 3 principles of how people learn. (p. 257)

1.5.   If each teacher sits down and asks these questions on a regular basis, each will be able to construct own set of answers, enrich our knowledge base, and improve mathematics teaching and learning for at least one group of students.  The teacher will also embody the essence of what it means to be a resourceful, self-regulating mathematics teacher. (p. 258)

2.      Deciding What Knowledge to Teach

2.1.   Deciding what to teach is made difficult by all the differing levels of the students and all the different standards in NCTM, state standards, curricula guides, etc.

2.2.   Many teachers decide what to teach by choosing one set of understandings that they want all students to acquire.

2.3.   Number sense is one of those understandings in the primary grades, especially since it will be used heavily in handling problems in other areas (algebra, geometry, stats, etc.).

2.4.   Number sense is easy to see and identify, but hard to define and teach. (p.259)

2.5.   Central conceptual structure for whole number has been found to be central to children’s mathematics learning and achievement in at least 2 ways:

2.5.1.      Enables children to make sense of a broad range of quantitative problems in a variety of contexts.

2.5.2.      It is the building block on which children learn more complex number concepts (e.g. double-digit numbers).

2.6.   By choosing number sense, teachers demonstrate an intuitive understanding of essential role of this knowledge network and importance of teaching a core set of ideas that lie at the heart of learning and competency in the discipline (principle 2). (p.261)

2.7.   Developmental principles that should be considered in building learning paths and networks of knowledge for the domain of whole numbers:

2.7.1.      Build upon children’s current knowledge.

2.7.2.      Follow the natural developmental progression (defined from research) when selecting new knowledge to be taught, and thus creating a developmentally appropriate learning path.  This will be the base for higher level knowledge networks.

2.7.3.      Make sure children consolidate one level of understanding before moving on to the next.

2.7.4.      Give children many opportunities to use number concepts in broad range of contexts and to learn the language used in these contexts to describe quantity. (p. 266)

2.8.   Central Conceptual Structure Hypothesis

2.8.1.      Powerful organizing knowledge network that is extremely broad in its range of application and that plays a central role in enabling individuals to master problems that the domain presents.

2.8.2.      Central implies:

2.8.2.1.That the structure is vital to successful performance on a range of tasks, ones that often transcend individual disciplinary boundaries.

2.8.2.2.That future learning in these tasks is dependent on the structure, which often forms the initial core around which the subsequent learning is organized.

2.8.3.      Test by Griffin and Chase

2.8.3.1.Two groups (control and treatment), each consisted of kindergarten students of an age that typically have acquired number sense but had yet to do so.

2.8.3.2.Each group given pretest to test understanding of whole number.

2.8.3.3.One group taught via Number Worlds curriculum (treatment), based on teaching the central conceptual structure of number sense, while the other group (control) received a variety of other forms of math instruction for the same 10 weeks. (p. 264)

2.8.3.4.Treatment group improved substantially in all test areas, far surpassing the control group. (p. 265)

3.      Building on Children’s Understandings

3.1.   Just like anything else, the number knowledge that children have as they enter preschool around age 4 varies.

3.2.   To get a picture of their understandings, a teacher can consider the knowledge children typically demonstrate between the ages of 4 and 8 when asked oral questions from the Number Knowledge test.

3.2.1.      Administered individually.

3.2.2.      Administered orally.

3.2.3.      Prompts to elicit reasoning.

3.2.4.      Tool or set of questions a teacher can use to get children’s conceptions about number and quantity and understand the children’s strategies to solve the problems they are posed.

3.2.5.      Provides a good picture of entry and exit knowledge.

3.2.6.      Provides a model for ongoing, formative assessment (assessment centered classrooms).

3.2.7.      Divided into 3 levels. (p. 267)  drawn from cognitive developmental tradition that are hierarchical in nature (i.e. level 0 is foundational for knowledge demonstrated at level 1)

3.2.8.      Ages associated with each level represent the midpoint in the 2-year age period where the knowledge is typically constructed/demonstrated.

3.2.9.      Age norms are age ranges where children in developed societies (typically middle income children) typically pass that level of the test.

3.2.10.  Thus, test provides a set of broad developmental milestones for the majority of U.S. children.  However, how this holds true for children from vastly different sociocultural groups still must be determined.

3.3.   Understandings of 4-Year-Olds (p. 270)

3.3.1.      Most have constructed an initial counting schema (well-organized knowledge network) that enables them to count verbally from 1 to 5, use the one-to-one correspondence rule, and use the cardinality rule (last number counted is the number in the set).

3.3.2.      Most have also constructed an initial quantity schema that gives them an intuitive understanding of relative amount and of transformations that change that amount. (p.272)

3.3.3.      Most can also use words to talk about these quantity relations and transformations.

3.3.4.      Most DO NOT use these schemas in a coordinated or integrated fashion, as if they are stored in a separate part of the mind.

3.4.   Understandings of 5-Year-Olds

3.4.1.      Children begin to solve problems involving single-digit numbers and quantities without having real objects available to count.  Usually happens between 5 and 6.

3.4.2.      Children behave as if they are using a “mental counting line” inside their heads or with their fingers.

3.4.3.      Children are demonstrating their awareness that the counting numbers refers to real world quantities and can be used in the absence of countable objects. (p. 273)

3.4.4.      Children are able to tell which of 2 single-digit numbers is bigger or smaller instead of using objects that are physically present.  This new competence implies the presence of a sophisticated understanding.

3.4.5.      Children successful with these items appear to know.

3.4.5.1.Numbers indicate quantity and therefore…

3.4.5.2.Numbers themselves have magnitude.

3.4.5.3.The word “bigger” or “more” is sensible in this context.

3.4.5.4.That the numbers 7 and 9 have fixed positions in the counting sequence.

3.4.5.5.That seven comes before nine when counting up.

3.4.5.6.Numbers later in the sequence indicate larger quantities.

3.4.5.7.And that 9 is therefore bigger than 7.

3.5.   Understanding 6-Year-Olds (p. 274)

3.5.1.      Scholars hypothesize that around age 5 – 6, as children’s knowledge of counting and quantity becomes more elaborated and differentiated it also becomes more integrated, eventually merging into a single knowledge network (central conceptual structure for whole-number, or mental counting line structure).  This is a blueprint showing the important pieces of knowledge children have acquired and how they are interrelated. (p. 275)

3.5.2.      Around age 6 – 7, supported by entry into formal schooling, children typically learn the written numerals.

3.5.3.      When written numeral understanding is linked to their conceptual understanding of number, children understand that numerals are symbols for number words, both as ordered “counting tags” and as indicators of set size (numeral cardinality).

3.6.   Understanding 7-Year-Olds

3.6.1.      Around age 7 – 8, in grade 2, children are able to solve the same sorts of problems using double-digit numbers. (p.277)

3.6.2.      Children are also able to mentally use 2 counting lines for single-digit problems like, “How many whole numbers are between 2 and six?” 

3.7.   Understanding 8-Year-Olds

3.7.1.      Around 8 years old, children are able to add double-digit numbers.

3.7.2.      Note that the strategies are different.  Children with a solid understanding of using benchmarking numbers and the base 10 system are the most successful.  (i.e. to answer 12 + 54, a student explains that 10 plus 54 is 64 and 64 plus 2 is 66, so the answer is 66.) (p.278)

3.7.3.      Other children will use less sophisticated strategies, such as vertical format mental addition, or counting up by 12 from 54 in fingers.

3.7.4.      Around 7 – 8, children’s central conceptual understandings become more elaborate and differentiated, allowing them to represent 2 distinct quantitative dimensions (tens and ones) in a coordinated fashion.

3.7.5.      This is a new structure called bidimensional central conceptual structure for number where children are able to understand place value and are also able to solve problems involving 2 quantitative dimensions across multiple contexts, including time, money, and math class.

4.      Acknowledging Teachers’ Conceptions and Partial Understandings

4.1.   The Number Knowledge test can also provide an opportunity for teachers to examine their own mathematical knowledge and to consider whether any of the partial understandings children demonstrate are ones they share as well. (p. 279)

4.2.   Three Insights:

4.2.1.      Insight #1: Math is not about numbers, but about quantity

4.2.1.1.Math is about quantity, and numbers express quantity.

4.2.1.2.Numbers acquire meaning for children when they recognize that each number refers to a particular quantity and that numbers provide a means of describing quantity and quantity transformations more precisely.

4.2.1.3.They realize that numbers are tools that can be used to describe, predict, and explain real-world quantities and quantity transactions, and it gives them a big boost in mastering and using the number system.

4.2.2.      Insight #2: Counting words is the crucial link between the world of quantity and the world of formal symbols

4.2.2.1.Numbers are presented orally as a set of counting words and graphically as a symbol system.

4.2.2.2.Provide opportunities for children to connect the symbol system and counting words, which are more familiar to them when entering the formal schooling system from their prior knowledge.

4.2.2.3.Take advantage of the prior knowledge of the counting words. (p. 280)

4.2.2.4.Children need ample opportunities to use the oral language system to make sense of quantitative problems and that they be introduced to the graphic equivalents of the system in this familiar context.

4.2.3.      Insight #3: Acquiring an understanding of number is a lengthy, step-by-step process

4.2.3.1.Each new understanding builds systematically and incrementally on previous understandings.

4.2.3.2.Hopefully, each lesson or series of lessons will help a child move up 1 level at a time in his/her understanding and deepen/consolidate each new understanding before moving on to the next, and thus gradually constructing a more sophisticated and higher level understanding.

4.2.3.3.Process takes time, and each student may move at his/her own pace.

5.      Revisiting Question 2: Defining the Knowledge that Should be Taught (p. 281)

5.1.   Teachers are required to teach a whole classroom of children, so they need a set of general learning objectives for each grade level appropriate for the range of children involved.

5.1.1.      All children in the class need to obtain the developmental milestones – the cetral conceptual structures for whole number.

5.1.2.      All children become familiar with major ways in which number are represented and talked about so they can recognize and make sense of number problems they encounter across contexts.

5.2.   Based on the previous section and the Number Knowledge test, here are learning goals for each grade level Pre-K to grade 2 which are in the reach of the majority of children at each grade level so they can reach developmental milestones.

5.2.1.      Pre-K – Children acquire well-developed counting and quantity schema.

5.2.2.      K – Children acquire well consolidated conceptual structure for single-digit numbers.

5.2.3.      1^st – Children link structure to formal symbol system and to construct more elaborated knowledge network this entails.

5.2.4.      2^nd – Children acquire bidimensional central conceptual structure for double-digit numbers that underlies a solid understanding of the base-ten system.

5.3.   These goals develop a “number sense” learning pathway – a sequence of learning objectives teachers can use to individualize instruction for children who are learning at a faster or slower rate.

6.      How Can This Knowledge Be Taught?:  The Case of Number Worlds

6.1.   Number Worlds is a program designed to teach whole-number concepts using the How People Learn principles. (p. 282)

6.2.   Designed to build specifically on children’s existing understandings (principle 1).

6.3.   Designed to help children construct new knowledge, both factual and conceptual, that is organized in a way to facilitate retrieval and application (principle 2).

6.4.   Designed to require and teach metacognitive strategies (principle 3).

6.5.   Design Principles

6.5.1.      #1 Exposing children to major forms of number representation

6.5.1.1.Numbers are represented in our culture in 5 major ways:

6.5.1.1.1.      Objects

6.5.1.1.2.      Dot set patterns

6.5.1.1.3.      Segments on a line

6.5.1.1.4.      Segments on a scale or bar graph (p. 283)

6.5.1.1.5.      Segments or points on a dial

6.5.1.2.Children familiar with these forms of representation and language used to talk about number in these contexts have an easier time making sense of number problems they encounter inside and outside of school.

6.5.1.3.Number Worlds provides a way these can be taught and illustrates a knowledge-centered classroom.

6.5.1.4.At each grade level, children explore 5 different lands that correspond to the aforementioned number representations. (p. 284)

6.5.1.4.1.      Object Land (real objects – pennies, fingers, etc. – into groups)

6.5.1.4.2.      Picture Land (dot patterns representing mathematical sets – dice for example) (p. 285)

6.5.1.4.2.1.This land provides a link between the world of movable objects and the world of abstract symbols.

6.5.1.4.2.2.Children in this stage must make a mental correspondence between 2 sets. (p. 285)

6.5.1.4.2.3.Numerals and tally marks are also part of this land.

6.5.1.4.2.4.Children gradually come to think of these patterns as forming the same sort of ordered series as do the number words themselves.

6.5.1.4.3.      Line Land

6.5.1.4.3.1.Representation of numbers along a line like those found on board games like Chutes and Ladders.

6.5.1.4.3.2.Language of distance.

6.5.1.4.3.3.Children learn that a number can refer to a place on a line AND number of moves along a line.

6.5.1.4.3.4.Children learn to treat physical addition and subtraction of objects as the equivalent to movement forward or backward along a line.  This is very important in the transition from small countable objects to abstract numbers and numerical operations.

6.5.1.4.3.5.Until children make this transition, they are unable to move from physical to mental operations with any insight.

6.5.1.4.4.      Sky Land

6.5.1.4.4.1.Representation of number with bar graphs and scales, such as thermometers.

6.5.1.4.4.2.Always used in a vertical direction, such that bigger numbers are higher up.

6.5.1.4.4.3.Makes convenient context for introducing children to the concept of using numbers as measure, as a way to keep track of continuous quantity in standard units.

6.5.1.4.4.4.Helps develop children’s intuition for the properties of systems.

6.5.1.4.5.      Circle Land

6.5.1.4.5.1.Representation of numbers using dials, such as sundials and clocks.

6.5.1.4.5.2.More sophisticated since they incorporate a cyclic quality possessed by certain real-world dimensions like time and natural rhythm of the seasons.

6.5.1.4.5.3.Children develop spatial intuitions that become the foundation for understanding many concepts in mathematics dealing with circular motion (pie charts, time, number bases). (p.287)

6.5.1.5.They are introduced in the order of difficulty, but the goal is to help children appreciate the equivalence of these forms of representation and the language used to talk about numbers in these contexts.

6.5.1.6.Children are encouraged to explore all lands and all number representations early in the school year by beginning with activities in each land that target lower-level knowledge objectives and proceeding throughout the year to activities in each land that target high-level objectives.

6.5.1.7.For adults, different representations are easily seen to be equivalent, but they can seem like completely different worlds to children.

6.5.1.8.Helping children construct an organized knowledge network in which these ideas are interconnected is a major goal of Number Worlds.

6.5.2.      #2 Providing Opportunities to link the “world of quantity” with the “world of counting numbers” and the “world of formal symbols”

6.5.2.1.Plus Pup – An Object Land Activity for PreK/K programs

6.5.2.1.1.      Provide opportunities for children to count a set of objects and identify how many there are.

6.5.2.1.2.      Provide opportunities to recognize that when one object is added, the size of the set increased by 1. (p.288)

6.5.2.1.3.      By allowing children to explore, a range of strategies will emerge.

6.5.2.1.4.      By prompting children with questions, the teacher can lead students to make sense of the quantity transaction by describing it in their own words.

6.5.2.1.5.      By playing, children come to realize they can use the counting numbers themselves, with or without their fingers.  The teacher can then move onto predictions and proofs. (p.289)

6.5.2.1.6.      By encouraging problem-solving and communication, children’s thinking is visible and ongoing assessment opportunities are provided.

6.5.2.1.7.      This game serves as a conceptual bridge between an increase in quantity in the real world, and the +1 symbol that describes this increase in formal mathematics.

6.5.2.2.Minus Mouse

6.5.2.2.1.      Introduced after Plus Pup (p. 290)

6.5.2.2.2.      Identical to Plus Pup, except the mouse takes one cookie away.

6.5.2.2.3.      Most young children prefer the comfort of the familiar and thrive on the opportunities the similarities provide for them to anticipate what happens, and confidently make predictions.

6.5.2.3.Plus Pup Meets Minus Mouse

6.5.2.3.1.      Make the problem more complex by adding Plus Pup and Minus Mouse to the same activity.  Either Plus Pup or Minus Mouse will surface based on a card drawn from a pile.

6.5.2.3.2.      The challenge is to interpret the icon with its associated symbol to determine the action that should be performed and to figure out how to solve the problem of how many cookies are in the bag.

6.5.2.3.3.      Scaffolds on the development of whole-number sense.

6.5.2.3.4.      In all of the activities, teachers can assess each child’s level of understanding based on the solutions constructed (or not) for each problem posed, the explanations provided, and the strategies employed.

6.5.2.3.5.      These assessments can provide the teaching with an appropriate starting place for the next activity (how many cookies in the bag) and the sorts of questions that should be posed to individual children to help them advance their knowledge. (p. 291)

6.5.3.      #3 Providing visual and spatial analogs of number representations that children can actively explore in a hands-on fashion

6.5.3.1.Spatial contexts created for the Number Worlds program often take the  form of game boards where numbers are depicted as positions on a line, scale, or dial and on which quantity is depicted as segments on these representations.

6.5.3.2.By using pawns to represent “self” and moving through these contexts to solve problems, children gain understanding of the relationship between moving along a line, scale, or dial and increases and decreases in quantity.

6.5.3.3.The Skating Party Game

6.5.3.3.1.      Played in Circle Land at the K level.

6.5.3.3.2.      Designed to help children realize that a dial is another device for representing quantity, and that the same relationships that apply between numbers and movement along a number line also apply to numbers and movement in this context.

6.5.3.3.3.      Game is a circular path that includes 10 segments (0-9) to help children understand the cyclical nature of the base-ten number system. 

6.5.3.3.4.      Objectives:

6.5.3.3.4.1.Identify or compute set size, and associate it with position on a dial.

6.5.3.3.4.2.Associate increasing quantity with moving around a dial.

6.5.3.3.4.3.Compare positions on dial to identify which have more, less, or the same amount.

6.5.3.3.4.4.Use this knowledge to solve a problem. (p. 292)

6.5.3.3.5.      Children take turns and roll a die and count the dots to move their pawns around the dial.  Award cards are collected each time they go around the dial and the one (or group) with the most cards wins.

6.5.3.3.6.      Questions posed during game play should be tuned to the child’s current level of understanding.

6.5.3.3.6.1.Who is furthest?

6.5.3.3.6.2.How much further do you have to go?

6.5.3.3.6.3.How do you know?

6.5.3.3.6.4.How come everyone had the same number of turns, but [name] has the most cards? (p. 293)

6.5.3.3.7.      There are some variations, such as using +1/0/-1 cards.

6.5.3.3.8.      Encourage children to construct their own answers by paying close attention to the activity. (p. 294)

6.5.3.3.9.      Children gradually build up their intuitive understanding of links among worlds of quantity (in spatial context), counting numbers, and formal symbols.

6.5.3.4.Rosemary’s Magic Shoes

6.5.3.4.1.      Developed for Line Land for 2^nd graders to help children build understanding of base-ten number system.

6.5.3.4.1.1.The prop is the Neighborhood Number Line, which is comprised of 10 blocks of houses, each containing 10 houses.

6.5.3.4.1.2.The character for this game is a monster-tracker name Rosemary who has a pair of magic shoes that allow her to leap over 10 houses in a single bound.

6.5.3.4.1.3.For the magic shoes to work, she must tell how many times to jump 10 houses and how many times to walk past 1 house.

6.5.3.4.1.4.Children take turns picking a number tile that indicates a house where the presence of a monster has been suspected.

6.5.3.4.1.5.Using the shoes, they move to the house as quickly and efficiently as possible, check for monsters, (by drawing a card), and place a sticker on it, showing it’s monster-free.  (p. 295)

6.5.3.4.1.6.In later versions, children are required to keep a record of RoseMary’s movements using the formal symbol system.

6.5.3.4.1.7.Children need to watch each other’s moves closely to see if other methods are more efficient and to share thinking with the class.

6.5.3.4.2.      Children realize that they can leap over 10 houses from any number sequence, not just decade markers.

6.5.3.4.3.      Children realize they don’t always need to move forward, as it might be more efficient to move to the closest 10’s marker and back a few steps.

6.5.3.4.4.      Children gain fluency in computing the distance between 2 numbers in the 1 – 100 sequence and in moving fluently from one location to the next, using benchmark values to do so.

6.5.3.4.5.      Children gain an appreciation of the relative values of numbers (92 is a long way from 9) and that 9 + 2 cannot be 92.

6.5.3.4.6.      This is an acquisition of fluency, factual knowledge, and conceptual understanding that are greatly facilitated by spatial analogs that can be explored in a hands-on fashion.

6.5.4.      #4 Engaging children’s emotions and capturing their imagination so knowledge constructed is embedded not only in their minds, but also in their hopes, fears, and passions

6.5.4.1.Dragon Quest

6.5.4.1.1.      Number Worlds program uses imagination in its activities, but this activity uses it to a greater extent.

6.5.4.1.2.      Addresses major learning goal for 1^st grade of helping children link their central conceptual structure for whole number to the formal number system. (p. 296)

6.5.4.1.3.      Developed for Picture Land in the 1^st grade program.

6.5.4.1.4.      Children must work most with numerals and operation signs.

6.5.4.1.4.1.The game is based around a fire-breathing dragon that is terrorizing their town.

6.5.4.1.4.2.The children are heroes chosen to put out the dragon’s fire.

6.5.4.1.4.3.The children need 10 (or 20 or more in later phases) pails of water, and if they enter the dragon’s area with less, they are taken prisoner and must be rescued by another player.

6.5.4.1.4.4.Children take turns rolling a die and moving along the colored game board where they can land on a well pile in which they draw a card from that indicates with images and symbols a certain number of pails of water.

6.5.4.1.4.5.The first child to reach the dragon’s layer with 10 pails of water can put out the dragon’s fire and free any prisoners.

6.5.4.1.5.      Children become capable of performing a series of successive addition and subtraction operations. (p. 297)

6.5.4.1.6.      This activity gives children the opportunity to use formal symbol system in more efficient ways and to make sense of quantitative problems they encounter in the course of their own activity.

6.5.5.      #5 Providing opportunities for children to acquire computational fluency as well as conceptual understanding

6.5.5.1.Computational fluency is given special attention in the warm-up period of each Number Worlds lesson.

6.5.5.2.In PreK and K, these activities typically take the form of count-up and count-down games played in each land with a prop appropriate for that land.

6.5.5.3.It makes it possible for children to acquire fluency in counting, and at the same time, to acquire conceptual understanding of the changes in quantity associated with each successive number (up or down).

6.5.5.4.Sky Land Blastoff

6.5.5.4.1.      Used after children have become reasonably fluent in count-up activity that uses the same prop.

6.5.5.4.2.      Children use a large, specially designed thermometer with a moveable red ribbon.

6.5.5.4.3.      Children pretend to be a rocket ship and count down while the teacher (or student) moves the ribbon.

6.5.5.4.4.      Seeing the “liquid” level drop gives children a good foundation for subtraction (p. 298) by allowing them to see that quantity decreases in scale height.

6.5.5.4.5.      Also lays the foundation for measurement.

6.5.5.4.6.       Complexity can be added by changing the position of the ribbon, or asking children or make redictions about where the ribbon will be if it is dropped X degrees.  In this way, the teacher exposes children to a learning path attuned to their understanding and allows them to develop a network of conceptual and procedural knowledge. (p. 299)

6.5.5.5.In programs for 1^st/2^nd grade, higher level computation skills are used in warm-ups, such as Guess My Number.

6.5.6.      #6 Encouraging the use of metacognitive processes (e.g. problem solving, communication, reasoning) that will facilitate knowledge construction

6.5.6.1.Additional supports are built into Number Worlds that support problem solving, communication, and reasoning.

6.5.6.1.1.      Question cards

6.5.6.1.1.1.Developed for specific stages of each small group game were designed to draw children’s attention to quantity displays they create during the game.

6.5.6.1.1.2.Prompts them to think about these quantities and describe them.

6.5.6.1.1.3.Prompts them to reflect on reasoning and put it into words.

6.5.6.1.1.4.Initially for use by the teacher, children can gradually take over the function, taking greater control over their own learning. (p. 300)

6.5.6.1.2.      Dialog prompts included in teacher’s guide.

6.5.6.1.2.1.Provides more general set of questions than those provided in the game.

6.5.6.1.2.2.Useful in prompting children to use metacognitive processes.

6.5.6.1.2.3.Do NOT give guidance on how a teacher should respond to answers children give at different age levels.

6.5.6.1.2.4.So in order to provide more/better follow-up questions and answers to children’s thinking, it is important that teachers have a better understanding of the answers that children give at these levels, and create increased opportunities for children to express their thinking to build experience and expertise.

6.5.6.1.3.      Wrap-Up periods provided at the end of each lesson.

6.5.6.1.3.1.Children assigned as “Reporters” for their problem-solving group describes the mathematical activity that the group did that day and what they learned.

6.5.6.1.3.2.The “Reporter” takes questions from the class, and any member of the team can help answer them.

6.5.6.1.3.3.Most significant portion of learning occurs here because children have the opportunity to reflect, explain concepts to their peers, and acquire more explicit understanding of the concepts.

6.5.6.2.With practice, teachers become increasingly skilled at asking good questions to get things going and taking a back seat.  It is a skill that takes practice and patience. (p. 301)

7.      What Sorts of Learning Does This Approach Make Possible?

7.1.   Number Worlds was created to address 3 major learning goals:

7.1.1.      Conceptual and procedural knowledge of number.

7.1.2.      Number sense.

7.1.3.      Interest in and positive attitude towards mathematics.

7.2.   Program evaluation has been focused on the extent to which children exposed to it have been able to demonstrate gains on any of these fronts.

7.2.1.      Groups of students followed have always been from low-income, predominantly inner-city communities.

7.2.2.      Assumption is that if it works well for them, it will work just as well, if not better for affluent communities.

7.2.3.      Several different forms of evaluation have been conducted.

7.2.3.1.Number Worlds program students vs. matched control group that had taken part in different math readiness programs.  Number Worlds students consistently outperformed control group on tests of mathematical knowledge, developmental measures, and experimental measures of learning potential.

7.2.3.2.Number World program student follow-up 1 year later (after completing K program) using a double-blind test.  Number Worlds students were again found to be superior in virtually all measures including teacher evaluations of “number sense”. (p. 302)

7.2.3.3.Number Worlds children tracked over a 3-year period vs. 2 other groups having superior/high level achievements in math.

7.2.3.3.1.      Number Worlds program students caught up with and gradually outstripped higher achieving students of the magnet school (acclaimed school with special math coordinator).

7.2.3.3.2.      Number Worlds program students outperformed the other group from end of K onward.

7.2.3.3.3.      Number Worlds program students compared favorably with students from China and Japan that were tested on the same measures.

7.2.4.      Findings provide clear evidence that programs based on the principles of How People Learn works for the population of children most in need of effective school-based instruction.

7.2.5.      Teachers and students using the Number Worlds program consistently report a positive attitude toward the teaching and learning of math. (p. 3003)

Source:
 
National Research Council of the National Academies.  (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press.  (Chapters Introduction, 5-8).

Chapter 5 – Mathematical Understanding: An Introduction (Karen C. Fuson, Mindy Kalchman, John D. Bransford

• RQs
  1. How do the principles of How People Learn apply to mathematics?
• Notes

1.      How Students Learn Mathematics

1.1.   Rarely taught with the 3 principles in mind.

1.2.   Currently students are learning mathematics as mastering procedures with no meaning behind it.

1.2.1.      Instead of connecting, and building upon, and refining mathematical understandings (principle 1), instruction overrides reasoning processes, replacing them with rules and procedures that disconnects problem solving from meaning making.

1.2.2.      Instead of organizing skills and competencies around core concepts (principle 2), the skills and competencies are the center of and/or whole of instruction.

1.2.3.      Procedural knowledge is divorced from decision making, students do not use metacognitive strategies (principle 3) when solving problems.  (p.217)

1.3.   “Mathematical Proficiency” instead of mastery of procedure, constituted by the following strands.  (AddingIt Up Report from the National Research Council)  These strands map to the principles.

1.3.1.      Conceptual Understanding – comprehension of mathematical concepts, operations, and relations.

1.3.2.      Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

1.3.3.      Strategic Competence ­– ability to formulate, represent, and solve mathematical problems.

1.3.4.      Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.

1.3.5.      Productive Disposition ­– habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy.  (p.218)

2.      Principle #1: Teachers must engage students preconceptions

2.1.   At a very early age, children begin to develop the awareness of number and it is universal though the rate differs depending on environmental influences.

2.2.   People possess resources in the form of informal strategy development and mathematical reasoning that can serve as a foundation for learning more abstract mathematics.  But the connections are not automatic, meaning there is no bridge between informal and formal mathematics.

2.2.1.      Brazillian street children can perform mathematics when making sales in the street, but were unable to answer similar problems when presented in a classroom context.

2.2.2.      Californian housewives are able to price compare and solve mathematical problems when shopping, but unable to solve similar problems presented abstractly in the classroom. 

2.2.3.      Men who handicapped horses could not apply the same skill to the stock market.

2.3.   Need to build on existing knowledge and engage students’ preconceptions, especially when they interfere with learning.

2.4.   Some preconceptions are fostered in early childhood and are very counterproductive and may lead to students believing mathematics is “not for them”. (p. 219)

2.4.1.      Preconception #1: Mathematics is about learning to compute.

2.4.1.1.Example: What, approximately, is 8/9 plus 12/13?

2.4.1.2.Most people want to find the common denominator, which makes the problem difficult, instead of just realizing that both numbers are close to one and thus the approximate answer is 2.

2.4.1.3.Mathematics is about problem solving and computation is a tool for use to that end when it is helpful.  This always “sense making” to take place.

2.4.2.      Preconception #2: Mathematics is about “following rules” to guarantee correct answers

2.4.2.1.  Mathematics is viewed as a cut-and-dried discipline that specifies rules for finding the right answers. (p. 220)

2.4.2.2.  In reality, mathematics is a constantly evolving field involving systematic pattern finding and continuing invention.

2.4.2.3.History of mathematics illustrates that what is taught in one point in time as a set of procedures really was a set of clever inventions designed to solve pervasive problems of everyday life.  (Examples: abacus and calculators)

2.4.2.4.If mathematics procedures are understood as inventions designed to make common problems more easily solvable, and to facilitate communications involving quantity, those procedures take on new meaning. 

2.4.3.      Preconception #3: Some people have to ability to “do math” and some don’t.

2.4.3.1.Very widespread in the United States, but not necessarily other countries.

2.4.3.2.Can easily become self-fulfilling prophecy. (p. 221)

2.4.3.3.The United States is more likely to assume that ability, not effort, is more important to learning mathematics, and that is socially acceptable not to put forth effort. 

2.4.3.4.Teachers in the United States are less likely than their counterparts to let students struggle for a while with problems, and simplify them so that the students don’t struggle at all.

2.4.3.5.Without conceptual understanding of the nature of problems and strategies for solving them, failure to retrieve learned procedures (rules forgotten after summer vacation) can leave a student completely at a loss.

2.4.3.6.Students also feel lost when they don’t “get it” from the beginning.

2.4.3.6.1.      If students learn to memorize procedures, they can be baffled by the unexplained. 

2.4.3.6.2.      Providing a secure conceptual understanding of mathematics that is linked to students’ sense-making capabilities is critical so that students can puzzle productively over new material, identify the source of their confusion, and ask questions when they do not understand.  (p. 222)

2.5.   Engaging students’ preconceptions and building on existing knowledge

2.5.1.      Perspective includes an understanding that the rules for computation and solution are a set of clever human inventions that in many cases allow us to solve complex problems more easily, and to communicate about those problems with each other more effectively and efficiently.

2.5.2.      How can we link formal mathematics training with students’ informal knowledge and problem-solving capacities?

2.5.3.      Features include:

2.5.3.1.Allowing students to use their own informal problem-solving strategies, at least initially, and then guiding their mathematical thinking to more effective strategies and advanced understandings.

2.5.3.2.Encouraging math talk so that students can clarify their strategies to themselves and others, and compare the benefits and limitations of alternate approaches.

2.5.3.3.Designing instructional activities that can effectively bridge commonly held misconceptions and targeted mathematical understandings.

2.5.4.      Allowing multiple strategies (p. 223)

2.5.4.1.If students believe there are multiple ways to solve a problem their engagement strategy is kept alive.

2.5.4.2.It does not mean that all strategies are equally good, but students can learn to evaluate different strategies for their advantages and disadvantages.

2.5.4.3.A wrong answer is usually partially correct and reflects some understanding, so finding the part that is wrong and understanding why it is wrong can be a powerful tool to understanding and promotes metacognitive competencies.

2.5.4.4.Understanding new methods can be worthwhile mathematical project for a class, and others can be involved in trying to figure out why the method works.

2.5.4.4.1.      Illustrates how a classroom community can function.

2.5.4.4.2.      Demonstrates that not all mathematical issues are solved or understood immediately; sometimes sustained work is necessary. (p. 224)

2.5.5.      Encouraging math talk

2.5.5.1.Students and teachers actively discuss how they approached various problems and why.

2.5.5.1.1.      Can help everyone in the classroom understand a given concept or method because it highlights contrasting approaches.

2.5.5.1.2.      Also facilitates other sorts of learning.

2.5.5.1.3.      Helps teachers become more learning focused and make stronger connections with each student.

2.5.5.1.4.      Can provide springboard for further instruction, enabling them to extend thinking more deeply of understand and correct errors.

2.5.5.2.Students need to feel comfortable expressing their ideas and revising their thinking when feedback suggests the need to do so.

2.5.5.3.Allows teachers to draw out and work with preconceptions students bring with them to the classroom and helps students learn how to do this sort of work for themselves and others.

2.5.5.4.Making drawings (and explanation of drawing) also helps and is a form of communication. 

2.5.5.4.1.      Can help students reapply concepts and methods.

2.5.5.4.2.      Can help students figure out problems when errors creep back into their methods.

2.5.5.5.Teachers need to use carefully designed visual, linguistic, and situational conceptual supports to help students connect their experiences to formal mathematical words, notations, and methods.  (p. 228)

2.5.5.5.1.      Modeling language and help students use it in their discussions.

2.5.5.5.2.      More advanced students help less advanced students to learn by modeling, asking questions, and helping others.  (p. 229)

2.5.6.      Designing bridging instructional activities

2.5.6.1.Research has uncovered common student preconceptions and points of difficulty with learning new mathematical concepts that can be addressed preemptively with carefully designed instructional activities.

2.5.6.2.A teacher or curriculum designer can make a framework for a math domain by selecting conceptual supports that will help students make links among math words, written notations, and quantities in that domain. 

2.5.6.3.Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students informal experiences and new formal mathematics.

3.      Principle #2: Understanding requires factual knowledge and conceptual frameworks

3.1.   Conceptual understanding and procedural fluency, as well as effective organization of knowledge that facilitates strategy development and adaptive reasoning.  (p. 231)

3.2.   Recognition in the weakness of conceptual understanding of students in the U.S. has resulted in increased attention to the problems involved in teaching mathematics as a set of procedural competencies.

3.3.   At the same time, students with too little procedural knowledge do not become competent and efficient problem solvers.

3.4.   Both factual and procedural knowledge are critical and must be closely linked, especially because mathematics will become more complex as a student progresses through the school years and students will need to build on those previous skills.

3.5.   Teacher’s challenge is to help students build and consolidate prerequisite competencies in a network of knowledge.  They also must provide sustained and spaced opportunities to consolidate new understandings and procedures.

3.6.   Networks of knowledge are organized as learning paths from informal concrete methods to abbreviated, more general, and more abstract methods. 

3.6.1.      Discussing multiple methods in class can help provide students a conceptual ladder that helps them move in a connected way from where they are to a more efficient, abstract approach.

3.6.2.      Students can adopt an intermediate procedure that they can explain so they are comfortable until they are able to move to the next step.

3.7.   Developing mathematical proficiency

3.7.1.      Requires students to master both concepts and procedural skills needed to reason and solve problems effectively in a particular domain.

3.7.2.      Deciding which advanced methods students should learn to attain proficiency is a policy matter involving judgment about how to use scarce instructional time.  (p. 232)

3.7.3.      Deciding methods must also take into account which methods are clearer conceptually and procedurally.

3.7.4.      Teachers do not always need to teach multiple methods, but they will arise frequently in the classroom, especially because students bring them from home or think differently about mathematical problems.  Discussing the different methods can help produce deeper understanding and flexibility.

3.7.5.      For less-advanced students, it might helpful to select an accessible method that can be understood and is efficient enough for the future, and for those students to concentrate on learning that method and be able to explain it.  It is possible to do this while facilitating problem-solving with alternate methods.

3.7.6.      Instruction to support mathematical proficiency

3.7.6.1.To support conceptual understanding and procedural fluency, primary concepts underlying an area of mathematics must be clear to the teacher.

3.7.6.2.Because math has been taught with emphasis on procedure, adults taught this way might have difficulty identifying or using core conceptual understandings in a math domain.  (p. 233)

3.7.6.3.Major shifts occur in whole numbers (quantity), rational numbers (proportion and relative number), and functions (dependence in quantitative relationships).  That is the basis of this book for mathematics.  (p. 234)

3.7.6.4.Balance must be maintained between learner-centered and knowledge-centered needs and must continue to relate to individual learner knowledge.  (p. 235)

4.      Principle #3: A metacognitive approach enables student self-monitoring.

4.1.   Learning about oneself as a learner, thinker, and problem solver is an important concept of metacognition.  (p. 236)

4.2.   Sometimes wisdom can not be simply parted from teacher to student.

4.3.   Students’ experiences have strong effects on their beliefs about themselves and their abilities to remember info and use it to solve problems.

4.4.   If a student is primarily frustrated, simply telling him he can do it will not have a great effect. 

4.5.   Instead, help student experience their own abilities and find patterns in problems, invent solutions, and contribute to and learn from discussions with others.

4.6.   For optimal learning, students need to reflect on their experiences and begin to see their ideas as instances of larger categories of ideas.

4.7.   “Number Sense”  (p. 238)

4.8.   Instruction that supports metacognition

4.8.1.      Making students’ thinking visible can be thought of as ongoing assessment.

4.8.2.      Assessment can include students so they become involved in thinking about their own mathematical progress.  This can be internalized as metacognitive self-monitoring. 

4.8.3.      Classroom communication about students’ mathematical thinking greatly facilitates both teacher and student assessment of learning.

4.8.3.1.Teachers can discern primitive solution methods.

4.8.3.2.Teachers can see particular difficulties.

4.8.3.3.Teachers can see how students are advancing in their helping and explaining abilities.

4.8.3.4.Students can learn some general problem solving strategies that apply to many different problems, such as “make a drawing” and “ask yourself questions” which are a means of self-monitoring.

4.8.3.5.Students can offer teachers a view into their thinking and provide info about how to better help along a learning path to efficient problem solving methods.

4.8.4.      Emphasis on debugging

4.8.4.1.Metacognitive functioning is also facilitated by shifting from a focus on answers that are right or wrong to a focus on “debugging” a wrong answer.

4.8.4.1.1.      Useful in complex problem solving (such as computer programming).

4.8.4.1.2.      Technological advances mean that more adults will need to do more complex problem solving and error identification.

4.8.4.1.3.      Good skill to have that can be learned in a mathematics classroom.

4.8.4.1.4.      Focus on understanding can help students debug their own errors.  (p. 238)

4.8.5.      Internal and external dialog as support for metacognition

4.8.5.1.Very important for students to communicate about mathematics and for teachers to help them learn to do so.

4.8.5.1.1.      Students can learn to reflect on and describe their mathematical thinking.

4.8.5.1.2.      They can learn to compare advantages and disadvantages of different methods of solving a problem.

4.8.5.1.3.      They can learn to ask thoughtful questions of other student thinking.

4.8.5.1.4.      They can learn to help each other in informal and formal ways.

4.8.5.2.Teachers must help students learn to interact properly and in a manner that is useful to the class.

4.8.5.3.The goal in all of this discussion is to advance everyone’s thinking and monitoring of their own understanding and that of other students.

4.8.6.      Seeking and giving help

4.8.6.1.Students must have enough confidence to not only to engage with problems and try to solve them, but also seek help when they are stuck.  (p. 241)

4.8.6.2.Students helping each other is helpful in developing metacognitive awareness of all parties.

4.9.   The Framework of How PeopleLearn: Seeking a balanced classroom environment

4.9.1.      Effective teaching and learning depend on balance between learner-centered, knowledge-centered, assessment-centered, and community-centered classroom environments.

4.9.2.      Overemphasis on learning-centered teaching results in insufficient attention to connection with valued knowledge networks, crucially important guiding roles of teachers and of learning accessible student methods, and the need to consolidate knowledge.

4.9.2.1.Some suggest that students invent all mathematical ideas and that we should wait until they do so rather than teach ideas. 

4.9.2.1.1.      Ignores the fact that all inventions are made within a supportive environment and that providing appropriate supports (scaffolding) can speed such inventions.

4.9.2.1.2.      Can hold students back and they need to be helped to more generalized “good enough” methods.

4.9.2.2.Balance means focus should be on sense making and understanding of the methods, not students inventing.  (p.242)

4.9.3.      Classroom discussions may not be sufficiently guided by the teacher through the learning path and go on aimlessly, and thus balance and direction is needed.

4.9.4.      Use of real-world situations and conceptual supports may not be connected enough to standard math notation.  Need a careful approach to the choices of core representations or bridging contexts that might guide students through a coherent learning path.

4.9.5.      Learning may not be consolidated enough because of excessive focus on initial learning activities.  Must make time for consolidation of learning with feedback loops (for errors) in order to attain mathematical fluency. 

5.      Teaching Communities

5.1.   The U.S. teaching style is still overwhelmingly traditional.  Making a move toward the principles in How People Learn can be aided by evaluating ones own teaching practices and participating in forms of teaching communities of like-minded colleagues.

5.1.1.      Can help in the creation of learning paths from traditional teacher to being more focused on conceptual understanding.

5.1.2.      Can organize discussions around issues that arise from teaching a curriculum that supports the conceptual manner.

5.1.3.      Other examples: video clubs, lesson studies.  (p.243)