"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein
Showing posts with label Spiral Approach. Show all posts
Showing posts with label Spiral Approach. Show all posts

Tuesday, September 16, 2014

A Curriculum Can Destroy Education

The previous posts on this blog have been emphasizing the role of teachers and resources in student learning. These are the avenues through which learning in schools can be improved. The curriculum is viewed simply as a wish list. Without proper implementation, it simply remains a wish list. Although a curriculum can not be expected to solve problems in basic education, a badly designed curriculum can exacerbate problems.

A long standing debate in education is content versus skills. This dichotomy is actually untrue for deep learning involves acquisition of both content and skills. An editorial in the Journal of Research in Science Teaching recently revisited what defines "meaningful learning". It starts by presenting the following figure (originally from Ege, Coppola, & Lawton, Journal of Chemical Education, 74, 74–83):

Above copied from Journal of Research in Science Teaching
Volume 51, Issue 6, pages 679-693, 12 JUL 2014 DOI: 10.1002/tea.21165
Content and skills are not opposite sides of a pole. These are two orthogonal axes of learning. Students with low content but high skills have very limited factual knowledge, "someone who knows how to think, but who has nothing to think about." These are the "intellectual amnesiacs". Students with high content but low skills are likewise unable to progress since these students have not been able to develop skills necessary to transfer what they have learned into a new or different area. These are the "encyclopedist learners". What we need are the "expert learners", which from the above diagram is clearly a product of emphasizing both content and skills. The editors of the Journal of Research in Science Teaching are quick to point out that the argument of analogical versus rote learning is likewise a false dichotomy:

Above copied from Journal of Research in Science Teaching
Volume 51, Issue 6, pages 679-693, 12 JUL 2014 DOI: 10.1002/tea.21165
Meaningful learning as described by the chemistry faculty of the University of Michigan in their 1997 J. Chem. Ed. paper is:


In this light, one can look at a curriculum and ask if a student is indeed given ample opportunities to learn both skills and content. It is through this perspective that one could ask whether a spiral curriculum for both mathematics and the sciences in high school is the right or wrong way to go. The last sentence from the above excerpt answers this question. It is immersion that is required not a spiral progression....



Wednesday, July 31, 2013

To Loop or Not to Loop: Evidence Based versus Anecdotes

Recognizing coherence and progression as important factors in effective education combined with the philosophy of taking into account where the students currently stand, it is quite tempting to suggest that teachers should be assigned to a class and stay with that group of students for a period of years. The practice of placing the same group of students with one teacher for more than one year is referred to in education as looping. I experienced this when I was in grade school. My teacher in Grade 5 also taught me in Grade 6. The entire class plus the teacher was therefore identical for two years. It was like a family, at least for two years. I liked the teacher so I had a positive experience with looping and obviously with two years in a row, that teacher knew a lot about us. Recently, at Georgetown, I had the rare opportunity of teaching one class of chemistry majors three of the eight semesters they spent in college. I taught these students General Chemistry and Physical Chemistry and since most of them opted for the Department's honors program, these students also enrolled in a graduate course that I instructed on Nuclear Magnetic Resonance Spectroscopy. Similar to my experience in elementary years, I also got to know these students quite well. 

Looping and Spiral

Saturday, May 25, 2013

Spiral Curriculum: When and How? Redundant versus Progressive?

Republic Act 10533 of the Philippines, otherwise known as the "Enhanced Basic Education Act of 2013", not only adds two years to basic education and reiterates universal kindergarten, but also prescribes the standards and guidelines the Department of Education must follow in developing curriculum. One item under this prescription is:
"The curriculum shall use the spiral progression approach to ensure mastery of knowledge and skills after each level."
The following is an example taken from a presentation given by Merle Tan, illustrating how chemistry is integrated into the new DepEd K+12 curriculum:



In the same presentation, it is also mentioned that "Science curriculum framework of high performing countries (Australia, Brunei, England, Finland, Japan, Taiwan, Thailand, Singapore, New Zealand, USA (3 states)) follow a spiral progression and integrated approach at least up to G9". The presentation, however, fails to cite that in Singapore, for example, "Teachers for early grades are trained and teach in either math and science or in languages and social studies, not all subjects." (Schools in Singapore may provide lessons for educators here, Cleveland.com). The presentation also does not mention the following observation highlighted by the US National Science Board's Science and Engineering Indicators 2002:
Analyses conducted in conjunction with TIMSS (Schmidt, McKnight, and Raizen 1997) documented that curriculum guides in the United States include more topics than is the international norm. Most other countries focus on a limited number of topics, and each topic is generally completed before a new one is introduced. In contrast, U.S. curriculums follow a "spiral" approach: topics are introduced in an elemental form in the early grades, then elaborated and extended in subsequent grades. One result of this is that U.S. curriculums are quite repetitive, because the same topic appears and reappears at several different grades. Another result is that topics are not presented in any great depth, giving the U.S. curriculum the appearance of being unfocused and shallow.
The above is summarized in the following figure:

http://www.nsf.gov/statistics/seind02/c1/fig01-14.htm
The spiral curriculum is in fact viewed as one of the problems of basic education in the United States. This is likewise emphasized in a study on curriculum coherence (J. CURRICULUM STUDIES, 2005, VOL. 37, NO. 5, 525–559) where the following table (for the physical sciences), illustrating coherence in curriculum in the top performing countries (Singapore, the Czech Republic, Japan, and Korea) and the lack thereof in the United States, is presented:

In the above table, the topics covered by curriculum in the top performing countries are enclosed. The US curriculum is redundant while those of the top performing countries are coherent. Comparing the chemistry curriculum of the top performing countries against the Philippines' DepEd K+12 curriculum, it is clear that countries like Singapore are already teaching atoms, ions and molecules to Grade 7 students, which makes sense since these are the fundamental concepts of chemistry.

To understand the very important yet subtle considerations behind designing a spiral curriculum, excerpts from the following book by Cathy Seeley may be of assistance:

http://books.google.com/books?id=6xvkJNWTErAC&source=gbs_navlinks_s

Spiral curriculum, when and how? These are in fact very important questions which can easily decide whether a curriculum will succeed or fail. First, for most countries including the top performing ones, the spiral curriculum is only applied up to middle school (Grade 8). The international exam, TIMSS, is given to students in Grades 4 and 8. Students from the US are only average among developed countries in the Grade 8 exam, suggesting that problems lie mainly in the later elementary years and middle school. In the top performing countries, the foundations of physics (forces, time, space and motion) are first introduced in Grade 5, while the fundamental building blocks of chemical knowledge (atoms, molecules and ions) are taught in Grade 7.  Although these topics are likewise covered in the US curriculum, a little bit about everything is also presented to children during these years. The US curriculum is quite diffused. The top performing countries pay attention to coherence in the curriculum. Perhaps, this is the reasoning behind less breadth. These countries choose to emphasize instead depth in the foundations of these science disciplines. Along this line, the sequence is very important. Chemistry is taught first with atoms, molecules and ions. This is one major characteristic that is lacking in the Philippines' DepEd K+12 curriculum.

Another significant difference between the science curriculum in DepEd's K+12 and those of the top performing countries is the obvious fact that the Philippines curriculum is two years behind. The integrated science approach adopted by the US and other countries stops at the end of middle school (Grade 8) while the Philippines expects to achieve this only at the end of Grade 10.

These differences between curricula of countries, however big, may still not be the explanation behind student learning outcomes. Human learning requires steps. We learn to walk before we run. Coherence in curriculum is therefore a must. Coherence in a curriculum can be a given with instructors who are specialized to teach a particular subject. A teacher who has an education degree specializing in chemistry, with or without a curriculum, would know what to teach first. This, in fact, is one major difference between teachers in Singapore and those in the United States. Teachers in Singapore, even in the elementary years, are subject experts. Teaching science in an integrated approach requires specific training. Drawing a curriculum that recognizes the hierarchical nature of topics within a discipline not only provides the conditions helpful to learning, but also facilitates the required teaching abilities. A spiral curriculum that deals with a mile wide range of topics on various disciplines requires too much from any teacher. A spiral progression approach must consider the resources available. There is no point in introducing a curriculum that cannot be possibly implemented correctly. There is wisdom in "Less is More"....







Friday, January 4, 2013

Keeping a Close Watch on Education Reform

Education reform requires research for its direction. Oftentimes, changes in education are purely sparked by an advocacy that is considered good in itself. Take, for example, "education for all". No one would really argue against that. When education reform takes the objective of "better learning", then guidance from research as well as proper assessment are both required. Various education reforms come and go, boldly promising dramatic improvements in learning, but in the end, yielding very little in uplifting basic education. These reforms are not free, some are very costly. Thus, in the end, only enterprises correctly positioned to help implement these reforms gain financially. Having a public school system try products regularly can be quite lucrative.

Research, unfortunately, is also expensive and time-consuming. When a proposed change seems promising, why should one wait for more rigorous and well-designed studies? Each year of waiting translates to a class of school children being deprived of a new method, perhaps a more promising new curriculum. The classroom, after all, is not like a laboratory filled with mice. These classrooms are filled with children whose future depends on the learning that is now taking place. But we do this in health care. Drugs need to pass through extremely stringent trials.

An elementary classroom in the Philippines
There is an amount of pragmatism required to tackle education reforms. Public school education requires money. There are clearly resources that are fundamentally required: teachers, classrooms, chairs, toilets, and learning materials (textbooks and supplies). It should be obvious that without these, no education reform is worth the experiment. Thus, it may seem that research on education reform is already a luxury for a country that cannot even meet the basic requirements of public schooling. However, not having the backup of good studies maybe more expensive than doing nothing especially when the reforms do harm to the current system.

Various interventions or education reforms have been implemented in developed countries like the United States with varying degrees of success or failure. A huge difference between a country like the US and a developing country like the Philippines is money although wasting public funds even in the US is really not a good idea. What a country could afford still makes a difference. For this reason, it is even more imperative for a country with limited resources to exercise greater caution in embracing or implementing education reforms. If a new program has not been vetted, there is a greater need for a regular assessment to check along the way if the desired outcomes are being achieved or not.

California is one of the largest economies in the world. Its 2012-13 budget on K-12 education alone is $38 billion (that is more than a trillion pesos). If one adds federal as well as private funds, this number rises to $68 billion (which is already significantly bigger than the annual budget of the Philippine government). California also faces challenges in public education and has instituted various reforms to improve basic education. One of its reforms is "algebra for all", in which all grade 8 students are encouraged to take algebra with the hope of improving performance in mathematics as well as learning higher-level disciplines. This is quite an interesting reform especially in the light of the Philippines' DepEd's K to 12 spiral curriculum in the mathematics. One important consideration in math learning is the sequential nature of the discipline. The question of how important prerequisites are comes into mind.

The "algebra for all" movement in California started in 2003 so there is ample data now to assess the impact of the program in math education. Jian-Hua Liang, Paul E. Heckman, and Jamal Abedi recently evaluated the program:

Downloaded from http://epa.sagepub.com/content/34/3/328.full
Their findings are summarized as follows:
  • Encouraging students to take algebra earlier does not lead to more students taking higher level mathematics subjects
  • Enrolling more students in algebra has little or no impact on student's learning success
  • Without sufficient preparation, taking algebra early does not benefit students
And I would like to highlight the following excerpt:
Changing classroom practice involves learning, especially among the educators who support students’ learning. Teachers will not change their practices as a result of only being given directions and information about required changes (Richardson, 1996). For example, motivation is critical to undertaking learning (Ryan & Deci, 2000), especially complex thinking and behavior. That is the learner, either student or teacher, has to desire to change and learn, especially if these changes require more than routine alterations in their thoughts and actions. With regard to classroom changes to encourage algebra learning, this means the teachers have to be involved, for example, in the creation of their classroom actions and the ways they think about those actions they are changing to encourage student learning, if policies like algebra for all are to be realized in classrooms....
There are obviously salient points drawn from this study that apply generally to education reform, points that should be reflected upon as DepEd's K to 12 is implemented in Philippine schools....


Friday, November 2, 2012

While the Philippines Moves to Spiral Approach, Missouri Does the Opposite

School districts in the state of Missouri are changing their science curriculum for Grades 6 to 8. The reform primarily changes science instruction from a spiral approach to a field-focus curriculum. The Philippines, on the other hand, with DepEd's K to 12 goes in the opposite direction. Without debating which direction is the correct one to take, both need to face the challenge of a major transition. Poor implementation of an education reform leads to failure even if the change is the correct prescription. A major part of the implementation is the transition stage, which is crucial for the success of the reform. It is therefore necessary to pay close attention to the transition process as this stage can easily lead to failure if not implemented correctly. Missouri's efforts are assisted by institutions of higher learning within the state. One is Lindenwood University.
The Spellman Clock Tower of Lindenwood University reflecting its view on education
One dissertation from Lindenwood University tackles specifically the transition of Missouri school districts to the new science curriculum: 
http://gradworks.umi.com/3450281.pdf
Abstract 
This investigation examined the transition from a spiral science curriculum to a field-focus science curriculum in middle school. A spiral science curriculum focuses on a small part of each field of science during each middle school year, more of a general science concept. In contrast to that, the base of a field-focus curriculum is that each grade level focuses on a specific field of science, more of a high school like concept. The literature reviewed provides a history of science education, the steps of the change process, and the importance of professional development. The literature review provided a basis for determining trends in the science education. 
The researcher collected a variety of data to understand the process that districts move through to transition to a field-focus science curriculum. Interviews provided information concerning the transition process of three Midwestern school districts that have arranged their curriculum into a field-focus alignment. Teacher surveys of one district supplied the perceptions of the professional development involved during the transition process. The researcher also examined school district student achievement data in the area of science. 
Suggestions made through this investigation focused on the Eight Steps to a Successful Change when implementing a field-focus science curriculum alignment. Following the suggested steps will help a transition go smoother.
This study specifically looks at the New Heart School District in the state of Missouri. The science teachers in this district have agreed to abandon the spiral approach and adopt a field-focus approach to teaching science. The rationale was simple - surrounding school districts that have instituted this reform are doing better in statewide standard exams. The following are among Alwardt's findings regarding the transition New Heart School District undertook:
  • Transition is always difficult so it is important that evidence supporting the reform is shared. In this particular case, data supporting the notion that a spiral approach leads only to a superficial treatment of topics and does not prepare students for the the rigor expected in standard tests.
  • Communication is vital between supervisors and teachers. These need to be regular so that updates and concerns are immediately addressed.
  • All necessary materials required for the new curriculum are promptly provided to all teachers. This effectively alleviates tension and anxiety toward the new curriculum.
As Alwardt emphasizes, "Transitions are inherently difficult for teachers." While trying to adjust to the change, teachers still have the obligation to give the very best instruction to the students. There are no "dress rehearsals". It is therefore very important that teachers during this stage are heard and supported. With these in mind, one can evaluate how DepEd in the Philippines is implementing its K to 12. One should understand and appreciate the crucial role of teachers in education reform.

Wednesday, September 26, 2012

Sequence of Science Courses

DepEd's K to 12 employs the spiral curriculum in teaching sciences in high school. For example, in grade 8, the first quarter is assigned to chemistry topics which include the particle nature of matter, atomic structure, and the periodic table. The second quarter is mostly biology dealing with a wide spectrum of topics; the digestive system, cell division, biodiversity, and ecosystems. Physics is studied during the third quarter and in this year, the areas discussed are the laws of motion, work, power, and the different forms of energy. The fourth quarter is on earth sciences which include earthquakes, typhoons and the solar system. Looking back at Grade 7, one may then evaluate what the sequence of topics is and ask whether the various disciplines maybe influencing each other. In chemistry, Grade 7 talks about solutions, acids and bases, elements and compounds, and metals and nonmetals. Biology in Grade 7 seems to prepare students for Grade 8 biology as it covers parts and functions, heredity, and interactions within an ecosystem. Physics likewise as it introduces force, motion and energy. And the last quarter deals with the climate in the Philippines, the atmosphere, and eclipses.

Whether there are cross-disciplinary benefits is an important question. This in fact is an active research area for education in the United States. In this light, the sequence may be relevant. The spiral curriculum could be regarded as an extreme design of mixing the sciences. Cross-disciplinary benefits are more likely to happen when a student covers one branch of science for an entire year. The spiral curriculum can only devote one quarter of a year to each branch, so the topics student will be exposed per year in each branch of science are severely limited. The following in a study that describes how chemistry, for example, may aid in learning biology. This is an abstract of an Honors Thesis submitted by Lauren Kronthal to the Department of Chemistry at Georgetown University in 2012:


A Background in Chemistry Helps Students
Learn and Understand Biology
Lauren J. Kronthal
Thesis Advisors:  Sarah Stoll, Ph.D. and Gina Wimp, Ph.D.
Abstract
         With the booming science, technology, engineering, and math job market, the United States cannot afford to be behind in the sciences if it is to remain economically competitive with other industrialized nations. High schools are desperately trying to improve their students’ understanding of the sciences by switching the order of science classes based on the suggestions of educational researchers. Recently, educators have proposed that chemistry be taught before biology since chemistry is necessary to fully understand biological concepts, but no empirical studies have been performed to show that chemistry improves student understanding of biology. I, therefore, addressed the question: Does a background in chemistry help students understand biological concepts?
            To address this question, I taught different biological concepts by 1) providing the relevant chemistry background or 2) not providing such background. I gave an assessment with questions of varying difficulty levels for topics where a chemistry background was provided/not provided and graded student responses. I found that a background in chemistry significantly improved students’ scores on questions that tested basic recall of information and on questions that required students to create a new idea using their knowledge of the content. Other levels of questions had no difference in mean class scores between when chemistry was taught and when it was not taught. Overall, students performed significantly better when given a background in chemistry. These results show that teaching chemistry before biology in high school can help improve student understanding of biological concepts.

To understand what the above study is really about, it is important to look at exactly what topics were being taught in chemistry and biology. The chemistry lectures are on intermolecular forces, polar and nonpolar compounds, and solutions, while the topics covered in biology are the sugars; monosaccharides, disaccharides and polysaccharides, as well, as movement of ions and water inside cells. In this case, the biology topics clearly benefit from a background in chemistry. Chemistry provides a perspective that allows students to see the components inside a cell in molecular terms. What is important in this curriculum design is a deliberate effort to connect the topics between the two fields of science. Such is not evident in the DepEd's K to 12 science curriculum.

The biggest disadvantage of a spiral curriculum is the lack opportunity to cover a variety of topics within one discipline in a year. Each discipline requires steps. To get to intermolecular forces and a molecular understanding of solutions, there are prerequisites. The topics build on top of each other and a quarter is simply not enough time to cover enough to aid the student in another field. It is simply the nature of the subject. Thus, designing a curriculum that will achieve what is described above will require a year of chemistry before taking biology.

Whether taking one subject in science helps in another is an important question. A survey of how students perform in college science courses provides preliminary insights:

Figure downloaded from  http://www.education.rec.ri.cmu.edu/roboticscurriculum/research/Sadler%20Tai.pdf 
The above does not directly answer the question since this is a study of how students performed in these fields after finishing high school. However, although it does not specifically address how a student's background affects a student's performance on a science subject in high school, it clearly shows that there are cross-subject benefits. Of special interest, is how high school math influences a student's performance in all sciences, including biology. The fact that students who had high school calculus perform much better across the board is probably not so much on an improvement in background, but more on being exposed to greater challenges. These studies are still ongoing and these illustrate how reforms in science education should be made. Reforms in science education can not be simply dictated in a whimsical fashion.  

Tuesday, September 25, 2012

Why Physics First: An Alternative to Spiral Curriculum in the Sciences


There is a movement regarding science education in high school in the United States that has been increasing in popularity. Spearheaded by a Nobel laureate in physics, Leon Lederman, "Physics First" makes the claim that the proper sequence for teaching the sciences in high school should be physics, followed by chemistry, and then biology. The project "American Renaissance in Science Education" summarizes this order in the following flow chart:
Figure downloaded from  http://ed.fnal.gov/arise/arise_lml/arise_science.html
The University of Missouri currently has a program that helps train teachers in implementing the above course sequence. It is briefly described in a brochure with the following brief rationale:


Downloaded from  http://www.physicsfirstmo.org/files/Brochure%20Aug09B.pdf 
Bottom line: Unlike the spiral curriculum that DepEd's K to 12 promotes, "Physics First" is a response to our improved understanding of how the brain learns. There are additional significant differences. "Physics First", as demonstrated in the University of Missouri program involves summer workshops for teachers over a three year period. This reform does not take place with teacher training lasting for a week or two. An institution of higher learning is intimately involved not for weeks, but for years. The program is not imposed on all public schools. And in the limited, well-designed, controlled studies, regular evaluation will be performed. This is in line with a perspective from another Nobel laureate, Richard Feynman:
"Anecdotal evidence alone, however, cannot confirm the success of the physics-first curriculum. Richard Feynman, renowned physicist and Nobel laureate, spoke of this lack of credible studies in science education almost 40 years ago. "There is an enormous number of studies and a great deal of statistics," he said in a speech about education at the Galileo Symposium in Italy in 1964, "…but they are mixtures of anecdotes, uncontrolled experiments, and very poorly controlled experiments, so that there is very little information as a result." Following this logic, the physics-first curriculum cannot be declared a complete success without well-controlled studies showing its utility in raising science literacy." 
From "Physics First in Science Education Reform" 
Vikram Pattanayak
Biochemistry and Biophysics, University of Pennsylvania

Friday, August 31, 2012

A Study of Math Curricula in the United States

Mathematica Policy Research, Inc., has been providing research and data collection to guide public policies. One of its ongoing efforts is to evaluate Mathematics curricula in the early grades. In the United States, there are several curricula and in this specific study, Mathematica chose to examine four: (1) Investigations in Number, Data and Space, (2) Math Expressions, (3) Saxon Math, and (4) Scott Foresman - Addison Wesley Mathematics. A summary describing each of these curricula can be obtained from this link: http://www.mathcurriculastudy.com/Curricula%20Summaries.pdf

The following are links to the websites of these four curricula:

(2) Math Expressions

Results of the study have been released to the public. These are described in the following link:
http://www.mathematica-mpr.com/Newsroom/Releases/2010/MathStudyYr2_11_10.asp and the formal report which include the data and analysis can be read in http://www.mathematica-mpr.com/publications/PDFs/education/mathcurricula_fstsndgrade.pdf.

There are statistically significant differences between the curricula and the findings favor "Math Expressions". I would like to highlight the following description of "Math Expressions" taken from the abstract obtained from http://www.mathcurriculastudy.com/Curricula%20Summaries.pdf (the bold emphasis is mine):

Key aspects of this curriculum include application of accessible algorithms that can be more easily understood and used by students; use of student math drawings and research-based visual representations to support student understanding and class discussion of mathematical thinking; an emphasis on in-depth sustained learning of core grade-level concepts (rather than a spiral curriculum) to support students’ conceptual understanding and fluency; and a “learn by teaching” design to support teachers new to the curriculum.
Contrast the above study with DepEd's Secretary Luistro's view:

"Mapapagbuti na natin ang kaalaman ng mga mag-aaral sa Agham at Matematika sa pagsusulong ng spiral approach sa ating bagong curriculum," the DepEd chief said. (With the spiral approach in the new curriculum, we will improve learning in math and science)

http://www.gmanetwork.com/news/story/256058/news/nation/deped-pinoy-school-kids-would-be-able-to-read-before-they-enter-grade-2-of-k-12

Tuesday, July 17, 2012

DepEd's Spiral Curriculum II


Browsing through DepEd's curriculum guide for science, one can pick from the grade level standards elements that are related to chemistry:
Grade 3: Students will learn that things may be solid, liquid or gas while others may give off light, heat and sound.  
Grade 4: After investigating, learners will identify materials that do not decay and use this knowledge to help minimize waste at home, school, and in the community. They will also investigate changes in the properties of materials when these are subjected to different conditions. 
Grade 5: After investigating, learners will decide whether materials are safe and useful based on their properties. They will also infer that new materials may form when there are changes in properties. Learners will recognize that different materials react differently with heat, light, and sound. They will relate these abilities of materials to their specific uses. 
Grade 6: Learners will recognize that when mixed together, materials do not form new ones thus these materials may be recovered using different separation techniques. Learners will also prepare useful mixtures such as food, drinks and herbal medicines. 
Grade 7: Learners will recognize the system of classification of matter through semi-guided investigations but emphasizing fair testing. 
Grade 8: Learners will explain the behavior of matter in terms of the particles it is made of. They will also recognize that ingredients in food and medical products are made up of these particles and are absorbed by the body in the form of ions. 
Grade 9: Learners will explain how new materials are formed when atoms are rearranged. They will also recognize that a wide variety of useful compounds may arise from such rearrangements. 
Grade 10: Learners will recognize the importance of controlling the conditions under which a phenomenon or reaction occurs. They will also recognize that cells and tissues of the human body are made up of water, a few kinds of ions, and biomolecules. These biomolecules may also be found in the food they eat.
Downloaded from  http://diylol.com/
To compare DepEd's K to 12 treatment of chemistry with the basic education curriculum in other countries, one can combine Grades 7 to 10. Each of these grades in DepEd's K to 12, assigns one quarter of the year to chemistry. Adding these through the first four years of high school sums up to one year of instruction in chemistry. This, of course, only allows for comparison on the basis of the length of time devoted to the subject. In countries where chemistry is taught as a year-long subject, there are no three-quarter gaps between each incursion into chemistry. The flow of concepts covered can be managed more easily in a year-long subject than in a spiral curriculum. The brief instructions in chemistry followed by long gaps require significant long term memory on the part of the students and it is likely that each year would require a significant amount of review of previous material. Nonetheless, even in the ideal scenario where students retain what they have learned in each year, there are huge differences between the topics covered by DepEd's K to 12 and those found in other countries. An example of what is generally covered in high school chemistry in the United States is provided by Dr. Anne Marie Helmenstein, PhD.  (See Topics Studied in High School Chemistry)

One of these differences is very important. The word "stoichiometry" cannot be found in DepEd's K to 12 curriculum guide. Tai, Ward and Sadler, in a study published in the Journal of Chemical Education ("High school chemistry content background of introductory college chemistry students and its association with college chemistry grades." J. Chem. Ed., 2006, 83(11), 1703-1711.), found that of all the topics that high school chemistry covers, only "stoichiometry" is found to be a good predictor of college chemistry performance. They arrived at this conclusion from a survey of more than 3000 students across the United States. The statistical analysis shows convincingly that performance in introductory courses in chemistry in college is strongly correlated with how well stoichiometry was covered in high school. And excerpts from individual responses from students provided a glimpse of the underlying reason behind this strong correlation:

I think stoichiometry gave a lot of kids trouble so I think my fairly strong background with that gave me a heads up. 
...stoichiometry—I learned that really well in high school and I remembered it all throughout chemistry. 
...knowledge about stoichiometry from high school chemistry helped me most. 
I’d have to say stoichiometry because quite a few people had problems with that.” 
...stoichiometry and the ability to apply conversions helped the most. 
...most helpful was the depth [with which] we covered stoichiometry....
What is stoichiometry? For the benefit of readers who do not have any background in chemistry, here is "stoichiometry" as described by Wikipedia:
Stoichiometry (play /ˌstɔɪkiˈɒmɨtri/) is a branch of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In a balanced chemical reaction, the relations among quantities of reactants and products typically form a ratio of whole numbers. For example, in a reaction that forms ammonia (NH3), exactly one molecule of nitrogen (N2) reacts with three molecules of hydrogen (H2) to produce two molecules of NH3:
N2 + 3H2 → 2NH3
Stoichiometry can be used to find quantities such as the amount of products (in mass, moles, volume, etc.) that can be produced with given reactants and percent yield (the percentage of the given reactant that is made into the product). Stoichiometry calculations can predict how elements and components diluted in a standard solution react in experimental conditions. Stoichiometry is founded on the law of conservation of mass: the mass of the reactants equals the mass of the products.
When I was teaching chemistry to non science majors at the Ateneo, I used the assembly of bicycles to illustrate stoichiometry. I begin with the assumption that all bicycle parts are available except for the handles and tires. I then ask the class how many bicycles I could assemble if I had two handles and two pairs of tires and of course, everyone answers "two". Interestingly, when I change the initial conditions to having 5 pairs of tires and only one handle, the class got the correct answer as well, only one bicycle, because I was limited by the number of handles I have. This is stoichiometry. In a chemical reaction, a given ratio needs to be met by the starting materials. If this ratio is not met, there will be an excess in one of the starting materials while another will be totally used. Doing this in chemistry involves arithmetic since substances react with each other in units of either ions, atoms or molecules. Thus, to evaluate the stoichiometry of a given reaction, one must first know how to convert quantities that we are familiar with in the macroscopic world such as mass or volume into units that are appropriate for the microscopic world of atoms, ions and molecules. Stoichiometry is quantitative and for this reason, in high schools where only the descriptive or qualitative aspects of chemistry are emphasized, this topic is often neglected. 
If stoichiometry is covered in college courses, why should its high school coverage have an effect on a student's performance especially when stoichiometry is only one of the more than two dozens of topics covered in an introductory course of chemistry in college? A college instructor should be able to address and teach stoichiometry. Herein lies one of the principles of basic education. Basic education not only teaches what to think, but also how to think. Basic education should make students think. And this is what students do when they perform calculations in stoichiometry. And if students fail to learn this in high school, they will have great difficulty not just in college, but in real life. We should not leave this important ingredient in chemistry education to the additional two years in high school. At that time, the two tracks are already in place. And not everyone will be taking chemistry as a subject.
DepEd's K to 12 chemistry fails to emphasize stoichiometry because it likewise neglects what chemistry is all about. Chemistry is founded on seeing the world through the eyes of atoms and molecules. Chemists see reactions in the following way:

Examples of Lewis acid-base equilibria.
Downloaded from 
http://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react1.htm
With the images above, it is easy to see why there are given proportions in chemical reactions. The combinations are as specific as how a bicycle is assembled. The arrows also go both ways, which denote equilibrium, another important concept in chemistry, which is likewise not mentioned in DepEd's K to 12 curriculum guide for science. Obviously, I have other fundamental objections to DepEd's curriculum in chemistry. I do not even know what DepEd means by saying, "They will also recognize that ingredients in food and medical products are made up of these particles and are absorbed by the body in the form of ions."  I can only scratch my head while reading this sentence.

Stoichiometry is indeed important. It illustrates a way of thinking and an important fact of life. Both teacher groups, the Alliance of Concerned Teachers and the Teachers' Dignity Coalition probably have their own version of stoichiometry as applied to improving Philippine basic education:

Unfortunately, President Aquino's government does not even have any one of these ingredients....

Saturday, July 7, 2012

DepEd K to 12 Modules, For Sale?


No, this blog is not selling modules for DepEd K to 12. Although, it is noteworthy to point out a comment that I recently read on the Facebook page of the Alliance of Concerned Teachers. The comment describes how some division offices are distributing the new materials. A new workbook apparently costs 160 pesos. Of this price 120 pesos go to the supplier or publisher and the remaining 40 pesos are divided among the teacher adviser (20 pesos), department chairman or head (10 pesos), office of the principal (5 pesos), and the remaining 5 pesos go to the cooperative that sells the workbooks. This is rumor, of course. The fact, however, is that the new curriculum, with its spiral approach, creates the need for new instructional materials. The unique character of the new curriculum makes it difficult to resort to already available teaching materials. For example, the modules shown in the figure above comprise the first quarter of the science subject in the new grade 7 of DepEd's K to 12. This coverage is quite different from grade 7 science subjects in other countries. It is different from the science subject that I took when I started high school. And this will be followed by three quarters on entirely different topics in biology, physics and earth science.

The word "Diversity" in the title is highly appropriate since this set of materials for the first quarter already covers a very wide array of topics and concepts in chemistry. It illustrates one of the dangers of the spiral approach. It easily lends to a "mile wide and an inch deep" coverage. As a result, students fail to master the necessary skills to progress from one level to the next. The desire to cover something complex at the beginning disregards the need to acquire basic skills and understand the fundamentals of a science discipline. One can browse any general chemistry textbook and see that the topics covered in these modules are found not near the beginning of the book, but in much later chapters. The following are the chapters, for example, of Chemistry: The Central Science, Brown, LeMay, High School Edition:

1 Introduction: Matter and Measurement
2 Atoms, Molecules, and Ions
3 Stoichiometry: Calculations with Chemical Formulas and Equations
4 Reactions and Solution Stoichiometry
5 Thermochemistry
6 Electronic Structure of Atoms
7 Periodic Properties of the Elements
8 Basic Concepts of Chemical Bonding
9 Molecular Geometry and Bonding Theories
10 Gases
11 Liquids and Intermolecular Forces
12 Solids and Modern Materials
13 Properties of Solutions
14 Chemical Kinetics
15 Chemical Equilibrium
16 Acid–Base Equilibria
17 Additional Aspects of Aqueous Equilibria
18 Chemistry of the Environment
19 Chemical Thermodynamics
20 Electrochemistry
21 Nuclear Chemistry
22 Chemistry of the Nonmetals
23 Transition Metals and Coordination Chemistry
24 The Chemistry of Life: Organic and Biological Chemistry

Chemistry likewise depends on basic concepts and notions provided by other disciplines. It relies on arithmetic and geometry. The foundations of chemistry rest on the laws of physics. For this reason, chemistry usually begins with a review of fundamental forces, work, energy, space, time and measurements. Then, it marches to what chemistry is all about: atoms and molecules. The above modules are clearly on the surface and does not provide the pupils an opportunity to be immersed in the discipline. This is akin to teaching students what prime numbers are without teaching them first what whole numbers are and how numbers are divided.

(By the way, if one of the cups in the figure above contains coffee with cream - this is not a solution. See http://www.newton.dep.anl.gov/askasci/gen99/gen99499.htm)

Wednesday, July 4, 2012

Inquiry Requires Immersion. Not a Spiral Approach


"Inquiry is in part a state of mind — that of inquisitiveness. Most young children are naturally curious. They care enough to ask “why” and “how” questions. But if adults dismiss their incessant questions as silly and uninteresting, students can lose this gift of curiosity. Visit any second-grade classroom and you will generally find a class bursting with energy and excitement, where children are eager to make new observations and try to figure things out. What a contrast with many eighth-grade classes, where the students so often seem bored and disengaged from learning and from school!"
Bruce Alberts 


The spiral approach of DepEd's K to 12 to learning math and the sciences and the absence of a formal science subject in the early years will lead to a poorer basic education in these fields. On top of these, DepEd also claims a transition to an inquiry-based teaching. We do not think in a spiral fashion. Inquiry, which in layman's terms suggests "getting to the bottom of things", requires immersion, not a smorgasbord.

The National Academies Press of the United States provides materials that guide education in the sciences. The following is an example.


The above book, a publication of the National Academies Press (http://www.nap.edu/catalog.php?record_id=9596) talks about how inquiry happens in two settings, in science, and inside a classroom. The book starts with the two examples given below. The example from science is taken from a geological study of the Pacific coast of North America: 

Radiocarbon evidence for extensive plate-boundary rupture about 300 years ago at the Cascadia subduction zone
ALAN R. NELSON*, BRIAN F. ATWATER, PETER T. BOBROWSKY§, LEE-ANN BRADLEY*, JOHN J. CLAGUE, GARY A. CARVER, MARK E. DARIENZO£, WENDY C. GRANT, HAROLD W. KRUEGERstar, RODGER SPARKS**, THOMAS W. STAFFORD JR & MINZE STUIVER
*US Geological Survey, MS 966, Box 25046, Denver,Colorado 80225, USA
Institute of Arctic and Alpine Research, CB 450,University of Colorado, Boulder, Colorado 80309-0450, USA
US Geological Survey at Department of Geological Sciences,Box 351310, University of Washington, Seattle,Washington 98195-1310, USA
§British Columbia Geological Survey Branch, 1810 Blanshard Street,Victoria, British Columbia V8V 1X4, Canada
Geological Survey of Canada, 100 West Render Street, Vancouver,British Columbia V6B 1R8, Canada
Department of Geology, Humboldt State University, Arcata,California 95521, USA
£Geology Department, Portland State University, Box 751, Portland, Oregon 97207, USA
starKrueger Enterprises, Inc., Geochron Laboratories Division, Cambridge, Massachusetts 02138, USA
**Rafter Radiocarbon Laboratory, Nuclear Sciences Group, Institute of Geological and Nuclear Sciences, Ltd, Box 31 312, Lower Hutt, New Zealand
Department of Geological Sciences and Quaternary Research Center, Box 351310, University of Washington, Seattle, Washington 98195-1310, USA
THE Cascadia subduction zone, a region of converging tectonic plates along the Pacific coast of North America, has a geological history of very large plate-boundary earthquakes1,2, but no such earthquakes have struck this region since Euro-American settlement about 150 years ago. Geophysical estimates of the moment magnitudes (M w) of the largest such earthquakes range from 8 (ref. 3) to 9 1/2: (ref. 4). Radiocarbon dating of earthquake-killed vegetation can set upper bounds on earthquake size by constraining the length of plate boundary that ruptured in individual earth-quakes. Such dating has shown that the most recent rupture, or series of ruptures, extended at least 55 km along the Washington coast within a period of a few decades about 300 years ago5. Here we report 85 new 14C ages, which suggest that this most recent rupture (or series) extended at least 900 km between southern British Columbia and northern California. By comparing the 14C ages with written records of the past 150 years, we conclude that a single magnitude 9 earthquake, or a series of lesser earthquakes, ruptured most of the length of the Cascadia subduction zone between the late 1600s and early 1800s, and probably in the early 1700s.

The example from the classroom takes place in an elementary school:
"Several of the children in Mrs. Graham’s fifth grade class were excited when they returned to their room after recess one fall day. They pulled their teacher over to a window, pointed outside, and said, “We noticed something about the trees on the playground. What’s wrong with them?” Mrs. Graham didn’t know what they were concerned about, so she said, “Show me what you mean.”
The students pointed to three trees growing side by side. One had lost all its leaves, the middle one had multicolored leaves — mostly yellow — and the third had lush, green leaves. The children said, “Why are those three trees different? They used to look the same, didn’t they?” Mrs. Graham didn’t know the answer."
And the story continues with the entire class spending more than three weeks trying to find answers to their question. This is inquiry. This is immersion, not a smorgasbord, not a spiral approach.

Jeannie Fulbright, author of Apologia's Elementary Science Series, wrote an article on the immersion, and I am sharing her article (with her kind permission) here:


Using the Immersion Approach in Science

Though many educators promote the spiral approach to education wherein a child is exposed over and over again to minute amounts of a variety of science topics, we believe there is a far better way.

The theory goes that we just want to “expose” the child to science at this age. Each year he is given a tad bit more information than was given the year before, thus spiraling upward. However, this approach supposes that young minds are incapable of understanding deeper science; and education is thus dumbed down. Sadly, this '”exposure” method has proved unsuccessful in the public and private schools as NCES (National Center for Education Statistics) statistic show American eighth graders (all having been trained under this method) are consistently less than 50% proficient in science. This data reveals this approach to be an inadequate methodology in education.

If we continually present children with scant and insufficient science, they will fail to develop a love for the subject. If the learning is skimpy, the subject seems monotonous. The child is simply scratching the surface of the amazing and fascinating information available in science. And, sadly, students taught in this way are led to believe they "know all about" a subject, when in reality the subject is so much richer than they were allowed to know or explore.

That is why we recommend that kids, even young kids, are given an in-depth, above their perceived grade level, exploration into each science topic. You, the educator, have the opportunity to abandon methods that don't work so that your students can learn in the ways that have been proven effective.

The immersion approach is the way everyone, even young kids, learn best. That is why we major in one field in college and take many classes in that field alone. If you immerse your child in one field of science for an entire year, they will develop a love for both that subject and a love for learning in general. When a child really knows a subject, they become an expert on it. They have a genuine knowledge and understanding that most high school children haven't been able to comprehend.

However, if they rush through several fields of science in one year, they will feel unknowledgeable and insecure about the information. And in fact, they are unknowledgeable. But imagine the benefit to your child when he is able to authentically converse with the botanist at the botanical gardens, intelligently discussing the dynamics and idiosyncrasies that are seen in the plants. This will delight both your student and others with conversation that is actually interesting and intelligent, occurring because you discarded the method of teaching to the test, and studied a subject to a degree that your child knew the inner workings of that subject. A child taught in this manner learns to love knowledge and develop confidence.

Additionally, a child that is focused on one subject through an entire year is being challenged mentally in ways that will develop his or her ability to think critically and retain complex information. This will actually benefit the child and give him an advantage on achievement tests. He will be able to make more intelligent inferences about the right answer on science questions, as God has created an orderly world that works very similarly throughout all matters of science. A child who has not been given the deeper, more profound information will not understand how the scientific world operates, and can not even guess the correct answer on standardized tests.

Yes, it is wise to spend an entire year on one field of science. And I believe you will find that you, your children and their test results will profit greatly from this method. Science will become a favorite subject as the student finally attains to a greater understanding of God's world and how it works. And when he learns about another field, he will be able to make comparisons and contrasts, thinking critically about the subject because of his strong foundation.


Thursday, June 14, 2012

Why So Many Elementary Students Aren’t Mastering Basic Math Facts

by Lynne Diligent
Remedial Math in College
A friend of mine teaches remedial math at the Community College level. We were discussing the problem of a number of students who never seem to have their addition facts mastered (much less their multiplication facts). He wrote:

“I remember as a young math teacher wondering how many hours of flash card drill it takes in the elementary grades to become fluent in the addition and multiplication facts. I could imagine ten minutes a day of actual flash card drill, five days a week, for 45 weeks in one grade, a total of 50 hours if I multiplied correctly, might be a reasonable guess. Surely it has been studied. Well, if it has been studied I have never seen any evidence of it in the last fifty years. I thought of that as probably pretty basic knowledge about the teaching of arithmetic.”


I’d estimate that at least 30 hours of drill, spread over a period of time would be required for an average child to learn addition facts, and an equal amount of time later on to learn multiplication facts. But there is NO classroom time provided for this in the math curriculum.

The problem here, speaking as a third-grade teacher of 8-and-9-year-old students for a decade, is that the elementary math curriculum (in America) is not structured to provide ANY time for drill such as he describes. Even when I was a child in the early 1960′s, we did not have drill of that type in elementary school. My mother worked on flash cards with me 10-15 minutes every day before I was allowed to play. I HATED every moment of it, but saw the value of it when I got into the working world in my 20′s and was using multiplication every day, knowing my multiplication tables by heart thanks to her efforts.

I’ve been discussing math teaching with other math teachers for several years now, and I find there are several trends I highly disagree with.



Trend 1: The amount of math homework has been cut in half from 24-30 problems nightly, to 12-15. I can only guess that this has come about from parents complaining about too much homework over the years. While I am in favor of not giving more homework than necessary, unfortunately, the current lighter homework often does not give sufficient practice in a certain type of problem for the students to be able to understand or master that type of problem. One or two examples of a certain type of problem are just not sufficient.
Houghton Mifflin Grade 3 Math Homework prior to 2007 
Houghton Mifflin NEW Amount of Grade 3 math homework, starting in 2007
(taken from end of a Grade 2 workbook)

Trend 2: Drill practice is considered “old-fashioned.” Never mind that the teacher can make drill practice into a fun lesson, just like any other type of lesson can be made fun by a a dedicated teacher. Without any drill, and without parents practicing or drilling children at home (such as the type of flash card practice my mother did with me as a child), many children are just NEVER mastering even the basic addition facts, let alone multiplication facts.


I no longer teach Grade 3; I am now a private tutor. Unfortunately, I am now running across a number of 14-year-olds who are using calculators to add 5 + 3, or 7 + 6, or 9 + 2. What’s even worse, THEIR TEACHERS LET THEM!!!! I personally think calculators should just be thrown out until about Grade 11, or whenever math involving higher functions on calculators is started. Prior to that time, they shouldn’t be allowed in school at all.


When I taught Grade 3, I made students show all of their work on their homework, including every carry number, and every cross-out for borrowing; I didn’t allow them to say, “I did it in my head.” (See photo above of example homework prior to 2007.) One reason for making students show all of their work (I had several reasons) is that I knew perfectly well many of them had calculators at home. However, even if they did their homework with calculators, they would have to redo it to mark all the carry numbers and borrowing cross-outs. This makes it better to just do it by hand in the first place. I then spent 30 minutes of my teaching time DAILY, going over these homework problems. It’s so satisfying to a teacher to hear, “Oh! Now I see my mistake!” It’s a big mistake for a teacher just to mark answers right or wrong, as students learn nothing from that.


Trend 3: (mostly at the high school level, I haven’t yet seen it appearing in middle schools, although I could be mistaken): Don’t instruct and explain, and then follow up with practice to master the skills. Instead, put students into groups, and let them see if they can “figure out themselves” how to do problems. Don’t give much feedback, but of course, students will have the same test as if you taught them the traditional way. (So the parents who can afford it get math tutors to do at home the job that the teacher should be doing; the parents who cannot afford tutors or understand the math themselves have children who completely fail math).

High school math class, with students working in groups. 
Trend 4: (has been around for at least 25 years): It doesn’t matter if children don’t master a unit. Just move through all the units, and the same units will be covered next year in a little more detail. If they still don’t get it, the same thing will happen the following year, and hopefully they will get it then. This idea has a name, which is called something like “spiraling.”


Even though I’ve never seen it, in the past couple of years I’ve become aware that “Singapore math” requires mastery of each math subject to a certain degree before moving on to the next math subject.


I think students would be far better served by having HALF the number of math topics (eliminating topics in Grade 2 such as Data, Graphing and Probablility; Congruent Shapes and Symmetry; etc.) and making sure they have mastered basic addition facts (by heart), addition and subtraction of two-digit numbers, and multiplication tables up to 5 (by heart) before moving into Grade 3. If parents don’t have time to drill children at home on these facts, then some time for it should be allowed in the school curriculum.


One of the major problems with the “spiral” math curriculum is that in every grade, limited and precious classroom math time is being wasted on unnecessary math concepts, given the age of the students. Those who have put the spiral curriculum together have moved math education from practical, daily skills to incorporating many advanced and unnecessary skills (for the age of the students). Many of these topics could be saved for higher grades (6-8) and students would arrive better prepared, and intellectually ready.

Some important topics, which are covered briefly in the curriculum, but to which little or no time is devoted to practice or mastery of these important life skills: making change for customers, knowing addition and multiplication tables by heart, knowing how to do the simplest operations without a calculator, being able to recognize a wrong answer when a wrong button has been pushed on a calculator, developing estimation skills, becoming competent in measurement and fractions (useful to every housewife in halving or doubling recipes on a daily basis).

Consider: Are we not cooking anymore in American society? Are we not hanging picture frames? Are we not doing any home repairs or improvements ourselves? Is there never a need to count back change? Does no one sew or do woodworking for pleasure anymore?


There is also a great disconnect in many classrooms between the material students are working on, and on knowing the reason for learning it. Instead of letting students feel that they are learning skills which can be useful to them NOW, so much time is wasted on learning concepts where the only use is for passing a test which seems useless to the child. Younger elementary children are mostly concrete learners, and they love and appreciate fun concrete tasks to work on.

Here are five examples of the types of things I feel should be eliminated from the Grade 2 curriculum (for seven-year-old students):

Grade 2 spending time learning to differentiate between types of quadrilaterals.
Grade 2 spends time working on composition of shapes viewed in 3D.
Grade 2 spends time learning about congruent shapes.
Grade 2 spends time on learning about lines of symmetry. 
Grade 2 learns about slides, flips, and turns of geometrical figures.
Are the Chinese or Indian students spending time on these things at age 7? I doubt it.

In my opinion, more time needs to be spent on mastery of basic life skills in the early elementary grades.

One last point about the spiral curriculum. Math educator Brian Rude feels that the spiral curriculum should not be thrown out entirely, but that the problems are caused from barely touching on subjects each time, instead of cutting a bit deeper, so that information is retained. He points out, however, that if cuts are too deep, that there is a danger of never having time to return to that subject, and students will also forget. He feels a balance between the two extremes is best.

-Lynne Diligent




___________________________________________________________


Singapore Mathematics Curriculum: An Example of Spiral Approach


"Singapore mathematics curriculum emphasizes the spiral approach based on Jerome Bruner's explanation on spiral curriculum. The idea of the spiral curriculum, according to Bruner - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.' Most people will not miss the idea of 'repeatedly' but may miss the subtle notion of 'building upon them' and 'until the student has grasped the full formal apparatus' of the target concept. In Singapore curriculum, addition is taught four times in Grade 1 (this is a core idea and they are new to it) - addition within 10, within 20, within 40 and within 100. Students get to revisit the idea of addition repeatedly but each time building on the strategies that they already had. When they add with 10, they count all and count on, perhaps with the use of concrete objects and drawings. Later, in addition within 20, they learn to make ten before adding, effectively acquiring the notion of place value. Later they progress to more formal approaches such as adding ones and adding tens in the formal algorithm. Thus, it is not mere a review of materials. It involves extension. In a similar way, multiplication of whole numbers is taught in grades one through four; addition and subtraction of fractions is taught in grades two through five; area of plane figures is taught in grades three through seven; solving equations is taught in grades seven through nine. As a result, in Singapore, Algebra is taught across grade levels in high school (grades seven through twelve). Thus, we do not have the practice of teaching Algebra, Geometry etc separately. They are all under the subject of mathematics. There is geometry in all grade levels."

-Dr. Yeap Ban Har, Director of curriculum and professional development at Pathlight School, an autism-oriented K-10 school in Singapore, and the principal of Marshall Cavendish Institute, a global teacher professional development institute.