Foundations for Instruction
On this page you will learn about the Cambridge IGCSE program. If you are not familiar with the Cambridge website, I would highly recommend you explore it a bit.
Before getting into the curriculum, it should be understood that Cambridge IGCSE does not create list of things to know, for students. Instead, from grades K through 12, all courses share a desired outcome; the development of young minds. All courses at all grade levels should create a young person that is Responsible, Innovative, Confident, Engaged, and Reflective. (Read more here.)
A student that is successful in Cambridge will have received an education, in a classical sense. They'll be challenging thinkers and problem solvers who are resourceful, have excellent perseverance and can communicate their thinking in writing (mathematically and in English), and verbally.
Cambridge IGCSE is a curriculum designed to give 9th and 10th grade students a well rounded educational foundation. There are many classes offered, click here to see a list. Many schools offer Biology, World History, English, and Mathematics Part 1 to 9th graders, then Chemistry, US History, English, and Mathematics part 2 to 10th graders. There are a series of tests called Extended and Core, that help students learn where they rank, if they're prepared to explore future academic challenges in certain subjects.
Here's how it works. Students take their IGCSE courses (how many they take, and how that is organized is decided by your school). At the end of the course the students take a test. The tests are written by Cambridge International, are administered in very controlled testing conditions, and then are sent back to England where they are graded by Cambridge trained instructors and professors.
When a student achieves a passing mark on an IGCSE examination, they've demonstrated preparedness for college level material. If they achieve a score of A or A-star on an Extended level examination, that student is quite talented in that subject area and should be encouraged to pursue that subject further.
Here we will only concern ourselves with the mathematics test, syllabus 0444. Be warned, while the syllabus looks sparse, it is written in a different style than say the Common Core standards. It is up to the instructor to determine what exactly is meant and expected of students. To do so, an instructor should be very familiar with past examinations, how they are graded, the writing style, and of course the syllabus itself. The aim of this site is to help you, the teacher, with those things, as well as provide you with materials for instruction!
Core versus Extended
Perhaps the easiest way for a US teacher to understand the difference between Core and Extended would be to say that Core is similar to on-level students, while Extended is advanced or honors level. All of the topics in Core are also in Extended, but many expectations of a student taking an Extended course are not expected of a student taking a Core course.
It is my belief that the Extended curriculum is accessible by the vast majority of students. The limiting factor is not intelligence, but student skills and motivation. However, depending on how your school has arranged their Cambridge program, you may have the option of deciding which test is taken by which students.
There are two tests for Core and two tests for Extended. The Core tests are called Paper 1 and Paper 3. The Extended tests are called Paper 2 and Paper 4. (Click here to see what the tests look like, how they're graded and so on.) A student wishing to demonstrate readiness for more advanced mathematics will not be able to do so on a Core series of examinations, they must take Extended. Students who wish to pursue science, engineering, or mathematics, should be enrolled in the Extended courses.
Generally speaking the first tests, Paper 1 and Paper 2, are more knowledge based, while Paper 3 and Paper 4 involve abstraction, problem solving, and application of knowledge. Cambridge assigns letter grades to the students based on their performance. A letter grade of C means that a student is ready for college level material. For example, here in the US they would be prepared for College Algebra or higher. A letter grade of A or A-star means that student is exceptionally talented in the subject and is ready for A-level course, or Advanced Placement. (At my school, since we adopted the IGCSE curriculum for 9th and 10th graders as an honors program, our AP scores are improved greatly, both in number of students taking the tests and percentage of those that pass the tests. In 2018 we were awarded the District of the Year by the College Board.)
In summary, the IGCSE program is designed to educate a student.
The purpose of this page is to introduce you to how the Cambridge syllabus is written, how the test determines and ranks student performance, and how the test is used to inform your future instruction.
By the time you've worked through the steps here, which should take about three total hours, you will be better prepared to design your own scope and sequence (or better understand the one provided here), and will better understand how the demands of IGCSE differ other curricula like Common Core.
1. To begin, download the most recent Cambridge IGCSE Mathematics Syllabus.
Pay special attention to 4.1, Syllabus Goals. These are the ideas you must keep in mind when teaching. They'll guide you through the paradigm shift (From U.S.-style education), you must execute in order to best prepare your students.
The syllabus is not written sequentially. That is, student success will be unlikely if you begin with item 1.1 and continue onto item 1.2, then 1.3 and so on until you've completed all 10 units. It is the Cambridge philosophy that the teacher must determine the best way to work through the syllabus. The syllabus is a reference of what is to be taught. A textbook is a resource to help you with instructional materials and learning materials for students. A teacher should not assume that the sequence of a textbook is tried and tested, but instead is written for what makes sense for the author(s).
Before diving into the content standards, read and make sense of the course goals and outcomes. You'll need to be able to complete the following statement (for your students): At the end of the course, my students will ___________________________.
2. Then, take some time to look through the syllabus and the level you'll be teaching (Core or Extended). Try to make sense of what the syllabus is describing and anticipate what information and skill would be a foundation and what would be an extension. As an exercise, order the content standards for unit 2 (Algebra). As you'll see, the numeric sequence does not align with conceptual development.
3. Pick a past Paper 2, and take the test. Do the problems as your students are expected. When finished, annotate the content item numbers required on each problem. This will help you gain an insight into how integrated the problems on the papers are. Now, keep in mind, Paper 2 is largely procedural requiring minimal problem solving skill. As such, it is less integrated than Paper 4.
4. Grade your test according to the mark scheme. If you've not attended an Introductory Training you may reference the "How to Grade IGCSE," video linked here (under development).
5. Read the Examiner's Report. Note how the examiner reports globally on what was learned about student performance. If there is a problem you did not get correct on the test, check to see if you made a common mistake as reported by the examiner.
If what has been done in class will not promote future success, then class was a waste of time!
For students to be successful in Cambridge Math, class itself needs to operate differently than in a traditional United States math class. The students need to take on a much greater responsibility and the teacher needs to let go. To empower students as independent critical thinkers, the role of the teacher must change from expert to coach, guide and cheerleader.
First Idea: Math is Not Steps
It is typically the situation where students believe that math is about the steps. They'll ignore all of the information about the concept and the reasons behind things, but will jump all over the process for a particular problem. Cambridge tests are too well written to allow this type of superficial learning to be rewarded.
This is readily apparent when I teach the first topic, square roots. Students jump all over process, and avoid concepts and ideas because they see no value in them. It is only after they learn that the steps are determined by the concepts and ideas that they find value in them.
As a teacher it is my job, and your role if you so choose, to break students of this habit. Through constant reminding, withholding steps, asking students to describe what parts of equations mean, and coaching, I get students to write out their ideas, not in complete sentences, but in short hand. A prime example is in the picture below.
The most important writing on this problem is "Sector - Δ," because that's what drives all of the numbers and calculation after. Vigilance in this regard, focusing on the why we do something more than what or how we do it, is one of the major keys to success. And while the teacher sets the group norm by rewarding students for doing this and redirecting the reluctant, it is also the job of the student to always push for the deeper understanding and not to latch onto quick fixes.
Second Idea: Lack of Information is NOT a Problem
The amount of information that a student has access to is incredible. Whether a struggling student seeks the information they need or not is entirely up to them. It matters not if you provide it for them during remediation because the problem is failure to integrate information, not lack of access to it.
Students need to integrate the things they are learning into their body of mathematical understanding. They need to apply new concepts to discover process, not just be taught procedure.
An example of this would be when teaching exponents and negative exponents come up. I teach that exponents are repeated multiplication, written in short hand. I also teach them that a negative sign means opposite. So, negative exponents are the opposite of repeatedly multiplication -- they're repeated division. All of the properties and "rules" that come after are consequences of this!
So when we summarize the properties, some people call them laws, of exponents, I make sure the students can explain why things work the way they do.
Third Idea: Kids are Smart ... Let them Solve Problems
Perseverance, confidence and persistence are three great qualities. I believe students develop these in Cambridge Math. However the staying power of freshmen is notoriously lacking. Often, when challenged with a problem they'll likely give up, saying, "I don't get it," and engage in something more entertaining.
The phrase, "I don't get it," has no place in any math classroom because it is a blanket statement, it's emotive and a complaint. There is no way to respond to the statement in a constructive way beyond suggesting that they need to be specific about what is causing the confusion. Not only will this force them to reflect and think instead of just react, it develops a huge component of problem solving ... thinking!
As a teacher, set up problems that are nuanced, layered and complex, but approachable. Encourage students to stay engaged, explaining that if the first time they really try something is on test-day, they're going to fail.
Also, don't let the students off the hook by confirming complex answers as right or wrong too soon. In fact, get them used to defending and understanding their solutions to the point where they no longer look to you for confirmation. They need to understand that math is right or wrong because of situation, application and process, not because a person or book says so!
Fourth Idea: If You Can't Say It, You Don't Know It
Memory and language are inseparable. Memories are stories and without the language of the story, there is no memory. A student that cannot articulate what they know will have little to no chance of retention. So carefully stating what is understood or intended, with accurate vocabulary and limited pronouns, is a great exercise that promotes retention. Students that understand this are far more likely to engage in the act of deliberately expressing their ideas or understanding.
Fifth Idea: Cover all the materials with time to spare, then practice
Cambridge math covers a wide variety of material. All of Algebra 1 and Geometry, most of Algebra 2, and foundations of Trigonometry, Probability and Statistics as they're taught here in the US are covered. When the students take their end of course examinations, they're likely to see a problem that involves many of the topics all at once, and without prompt or warning that one of the topics is to be used.
In response to the nature of Cambridge exams I teach all of the subjects completely in the first 6.5 quarters of the program. That leaves me about 2 months of time to practice, review and shore up misconceptions and hit small things that always appear on the examinations.
The shift from expert disseminating information to eagerly receptive students to a mentor/coach that guides and encourages students through the education process is difficult.
The Cambridge philosophy is that all subjects, at all grade levels, have the same purpose. That is to develop an educated person. Your particular subject is a vehicle. Mastery of the subject is not the destination, but serves as a conduit to the final destination.
- If a student's learning is bound by the crafting of your lesson, they'll not learn much. Through questioning, by both teacher and students, a lot of unanticipated learning will occur.
- The purpose of an education is to develop a well-rounded, thoughtful individual that is a critical thinker, able to adapt their existing knowledge and incorporate new understanding in order to succeed at unanticipated tasks in the future. An educated person has learned perseverance, knows how to manage themselves in order to fulfill their potential and is articulate and considerate of various points of view.
- Your role, as a teacher, is to devise experiences relevant to your content area that promote the development of the attributes of an educated person. Your expertise in the content area is best applied in developing those experiences and guiding students through them, to help the students realize the intended outcome.
- Consider your subject a vehicle whose destination is an education for the student.
- A Cambridge student should be Responsible, Innovative, Confident, Engaged and Reflective. Our lessons should promote the development of these attributes.
In the video below you will be challenged to let go of student learning, to change your role from expert to cheerleader. Students learn what they want and they learn together. The more you can entice their "want," encourage them and allow them to explore and self-monitor, the better they'll learn.
While that may sound lofty, exploring some of the information on implementation will help you get some concrete things to try and places to start.
Policies and Procedures
Here are some of the policies and procedures that are aligned with the Cambridge philosophy. Some of these are in direct conflict with best practices here in the United States. Some you may disagree with, while other policies or procedures may be in violation of your district’s policies.
It may be a good idea to have a meeting with your administration team about the policies, their purpose, and how you plan to address the needs of your students.
The learning that occurs in a Cambridge classroom is instigated by the lesson. There are not a list of facts to know, or skills to possess. Watching a video will fail to create the cognitive dissonance required for learning. By exploring confusion through conversation, exploration and questioning, often in small student groups, learning occurs in class. This is likely impossible to replicate outside of the classroom environment.
This makes it extremely difficult for a student who is frequently absent to be successful in a Cambridge environment. They may be better suited in a slower-paced class.
Content and Language Objectives
Cambridge International does not believe in the long term efficacy of content and language objectives being posted for students to read. A typical objective might read, SWBAT solve quadratic equations by using the quadratic formula.
This is way too low of an expectation in a Cambridge class. In a Cambridge IGCSE math class students will be uncovering knowledge and applying new information in unanticipated ways. A posted objective that lays out what students will be doing will provide them with the path to a solution. This will not help them learn to be problem solvers. We want students to use the clues in the material to solve problems, not clues from the classroom itself.
If the posting of objectives is required, reference the syllabus and borrow their language. Have a discussion with your administration team about this.
Homework is an essential part of learning. Skill and Drill style worksheets are of little use because they only focus on procedure. Students never have to think about the procedure, when to apply it, or why it is useful.
Homework assignments should be of high quality, but not too time consuming. A high quality homework assignment forces students to challenge what they understand from class, to apply their understanding in a new way, and will expose common misconceptions.
Homework, when possible, should remain in the students’ notebooks. Students should write questions on their homework and make corrections beside their mistakes. This will prove to be a valuable studying resource for them.
Assign full credit for homework that shows a legitimate attempt at all problems. Encourage students to attend tutoring and to be resourceful when they are stuck on homework problems.
If a student is behind on their work, and trying to complete old assignments, two things are happening. First, the student has not had the learning experiences required to be successful with current topics. Second, instead of working with the current topics to prepare for how they’re applied, they’re focusing on old material. Most likely they are trying to get the late work done out of compliance and little learning benefit will be realized.
Yet, sometimes kids forget, sometimes big things happen in their lives. They do fall behind. It has proven to be a good policy to allow late work, with a penalty, so long as it is turned in the same week it was assigned.
Quizzes and tests are precious. Students show you what they’ve learned, which informs you and the student. It is imperative that assessments are graded quickly and returned with feedback to students. This allows you to stay in tune with student needs and provides them instant feedback on progress and issues that require remediation. Students should keep old assessments, they should write notes and corrections on the tests and keep them as a study resource.
Test retakes are generally a bad idea. A test retake policy that allows students to retake tests frequently hampers their development as students. The urgency they feel to perform well will motivate them to study. If they’re given multiple chances, this urgency is relieved.
If retests are allowed, it should be done with the idea of giving a student a second chance to demonstrate their knowledge and ability, not a way to improve their grade. This means that giving the same test, or just slightly modifying the original test is a horrible practice.
A test needs a significant portion of problems requiring problem solving. Problem solving is not the calculation, but making sense of the information provided contextually and devising a plan to arrive at a solution. Once a student has seen a question like this, even if they were unsuccessful in figuring it out, the question is no longer serving the purpose of measuring a student’s problem solving abilities.
Tests should be timed and graded in accordance with the style of Cambridge.
It is a good idea to define grades for students, administration and parents.
The letter grade A is awarded for mastery. This means the student could teach the material to another student in way that develops conceptual understanding in the other student. The earmark of mastery is the ability to communicate.
The letter grade B is awarded for fluency. Fluency is highly efficient, but there are small conceptual or procedural inefficiencies. Over time these issues compound.
The letter grade of C is awarded for proficiency. A proficient student possess enough conceptual understanding and procedural fluency to fulfill the standard. For demanding standards like those set by Cambridge IGCSE, proficiency is difficult for most students.
The letter grade of D is developing towards proficiency. The student has ideas, but has conceptual misconceptions inappropriate for the progress made by the class. With some remediation they can be caught up.
The letter grade of F is awarded for students that are not making sufficient progress. They would be better served in a slower paced course.
Parents may worry about a low GPA and college entrance. Students are fully capable of earning high grades in IGCSE, but the adjustment takes a little time. If your school calculates GPA on a semester basis, that is enough time! Because IGCSE is a college preparedness program, in earning a good grade students develop the exact skills required to be successful in college. Over time their grades will improve as their student skills increase. When they take the ACT or SAT test, their scores will be superb. It is often the case that IGCSE students test out of all required math courses for college!