:D

e-learning for Matrices - TLS

The e-learning for Matrices can be found in Blackboard under discussion board. We can continue to discuss the learning and teaching of Matrices here.

Math Modelling Survey - TLS

Can you all please take some time to fill out the following survey? Thanks so much.

https://spreadsheets.google.com/viewform?formkey=dDQybXFVRXM1a0oxb2MzOHdSZGNZOWc6MQ

The Monty Hall Problem on Youtube

Monty Hall Problem

Hey guys,

Here's there link to what I was talking about.

http://mathforum.org/dr/math/faq/faq.monty.hall.html

Scroll down to this,

"If you're still not convinced that 2/3 is the correct probability, here are two more ways to think about the problem.

1. It seems to make sense that you have a 1/3 chance of picking the correct door. This means, however, that since the probabilities must add up to one - and the car has to be somewhere - you also have a 2/3 chance of not picking the correct door. In other words, you are more likely not to win the car than to win it. Imagine that Monty opens a door and shows that there's only a goat behind it. Consider that the car is more likely to be behind a door other than the one you choose. Monty has just shown that one of those two doors - which together have the greater probability of concealing the car - actually conceals a goat. This means that you should definitely switch doors, because the remaining door now has a 2/3 chance of concealing the car. Why? Well, your first choice still has a 1/3 probability of being the correct door, so the additional 2/3 probability must be somewhere else. Since you know that one of the two doors that previously shared the 2/3 probability does not hide the car, you should switch to the other door, which still has a 2/3 chance of concealing the car.
   
2. What if there were 1,000 doors? You would have a 1/1,000 chance of picking the correct door. If Monty opens 998 doors, all of them with goats behind them, the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Here switching sounds like a pretty good idea."

Also, check out the next post for visual aids.

Merianna

Math Modelling Experience

I just want to share my math modelling experience that I had with contract school. Schools usually don't use the term 'modelling' but 'performance task'. Typically, schools integrate real world contexts into performance tasks. I rememeber doing one performance task with the secondary one students on floor tiling. Given a fixed floor area and 2 different types of tiles of different prices, the students were told to tile the floor based on a few criteria: optimum cost, least tiles wasted and least cutting of tiles involved. This kind of task really set the students thinking and at the end of the day, there is no fixed solution. But however, before the students even start to attempt the task, they will ask questions like 'is this counted in the examination?' and blah blah blah. I feel that in order to really engage the students' thinking in math modelling, the tone must first be set right; if not, you will have students giving superficial solutions, which you know they have not given much thought to it.

Another issue that arises is the evalution process, aka, rubrics. Does the rubric really provide the best solution? How do we assess the students? If the students fail in the performance task, does it mean that their ability is low? How do we train the students in mathematical modelling? These questions really pose some challenges in setting a performance task.

Well, I just feel that it may even take ages to come up with a good performance task with rubric that allows the teacher to assess the students' learning accordingly.

Group F

Something interesting to think about for educators out there..

Hello guys,

not sure if u have seen this video on FB recently. This guy was sharing about his opinion of the education system and the need to reform it (using animation to depict his ideas). Pretty thought provoking and the animation is amazing as well. Take a look.

http://www.youtube.com/watch?v=zDZFcDGpL4U

Modelling....

The modelling activity really brings back my university memories... I think it is good to let students experience such modelling techniques as I am sure it will be very helpful to them especially if they were to enter engineering or research areas in future where modelling becomes part and parcel of their lives...

As I reflect back regarding the part on using "square errors" rather than "absolute errors", I do not agree totally. I guess it really depends on situations. From what I learnt in process control, we term them as "Integral square errors (ISE)" and "Integral absolute errors (IAE)" respectively (it is actually the same as summation of the square errors etc since integral is basically summing up the area under the curve). In cases where large errors are particularly unacceptable, we will use ISE so as to amplify such errors otherwise, IAE is also often used. Nevertheless, in most journal papers on process control, engineers usually used both ISE and IAE to illustrate their results. 

I have also shared my old lecture notes - Classification of process models as follows (feel free to download it, it shows the various types of models as well as difference between them) :

http://www.mediafire.com/?akmuzukd1jrgd20 

Just some acronym used:

CSTR: Continuous stirred tank reactor

ODE: Ordinary differential equation

PDE: Partial differential equation 

Group A

Mathematical Modelling

Here is a website with a lot of resources on mathematical modelling. (http://www.math.montana.edu/frankw/ccp/modeling/topic.htm) I hope it is the same "modelling" that we are talking about in class.

Personally I find it is quite similar to steps in scientific methodologies. If you do have a graphical calculator or a data logger, you can try out some of the activities there.

Anyway enjoy reading!

Are areas made of parallel line segments?

I am not sure how many of you are convinced by the idea that the area is made of parallel line segments while the volume is made of parallel planes. Personally I am a little confused by this.

First of all, Cavalieri's principle does not implies this result. The statement of the principle is that "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal" (http://mathworld.wolfram.com/CavalierisPrinciple.html). Personally I think it is only looking at the cross-sections, but it does not imply that the solids are made of the thin slices. Similar argument applies to the area as well.

Regarding the activity that the parallel line segments shade area, personally I find it is a bit flawed. The line segments we have drawn in this activity, are not really line segments but thin rectangle blocks. If we are looking at the line segment itself, by applying a similar argument as the density of the real numbers, we can show that between any pairs of parallel line segments we have drawn, we can always find a space between these two line segments. Thus we cannot completely fill the area of the rectangle block with parallel line segments. Similarly in the activity with A4 papers, we have already assumed that each A4 paper has a specific volume. Thus we can then sum up the volume.

I am not sure how helpful it would be to the students at their age. However, I just find it would be a bit dangerous to let them carry this idea with them especially when they are working on problems at a higher level. Till now, I finally understand why some students have some difficulties in differentiating between projections and cross-sections, between point-groups and space-groups, when they are working with 3D chemical models.

Nevertheless, Cavalieri's principle is quite useful in deriving area or volume formula of figures/solids with special shapes. I have tried to look for a formal proof for this principle but so far I cannot find any. Shall we accept it as an inductive observation or a hypothesis? :)

Group A

Mensuration is covered this week and we were introduce to the Cavalieri's principle. In school, teachers usually do not explain how the formulas are derived or related to students as it is perceived to be too difficult for them to understand. Many times, students were just told to memorise the formulas and apply straight. It is good that we are introduce to Cavalieri's principle and we will finally be able to answer students when they ask how did all the formulas comes about. Though I agree that the principle is hard to them to understand and appreciate at their level, it is still good to bring it in as an introduction to the topic. Perhaps more 3D animations may help in enhancing understanding.

I found 2 sites that provide some java animations on the Cavalieri's principle. With it, students no longer have to try hard to visualise the diagrams, which I had a hard time too.

http://www.jimloy.com/cindy/cavalier.htm

http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.html

KSEG Free Interactive Geometry Software

KSEG is a Free (GPL) interactive geometry program for exploring Euclidean geometry. It runs on Unix-based platforms (according to users, it also compiles and runs on Mac OS X and should run on anything that Qt supports). You create a construction, such as a triangle with a circumcenter, and then, as you drag verteces of the triangle, you can see the circumcenter moving in real time. Of course, you can do a lot more than that--see the feature list below.
KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX. It is very fast, stable, and the UI has been designed for efficiency and consistency. I can usually make a construction in KSEG in less than half the time it takes me to do it with similar programs. Despite the name, it is Qt based and does not require KDE to run.

KSEG was inspired by the Geometer's Sketchpad, but it goes beyond the functionality that Sketchpad provides.

http://www.mit.edu/~ibaran/kseg.html

Group D

http://www.youtube.com/watch?v=Wv65HBdACEs

Hi All,

Here is something interesting to see. It is related to (but not exactly the same as) what we did today in tutorial.

Thanks,
Group D

Alternative proof for gradient of perpendicular lines

I thought the proof m1.m2 = -1 for gradient between perpendicular lines we discussed last week may be too difficult for lower abililty students.

Did some googling for an alternative. We could use simple diagram and rotate the line 90 degrees to demostrate the gradient relationship.



After rotating 90 degrees, from the diagram, the gradient becomes - B/A (vertical rise / horizontal run).
Is this way of proving good enough?

Reference: http://mathforum.org/library/drmath/view/54496.html

Group F

Reflection on Geometry Tutorial by Group F

Last week, we spent lots of time both during tutorial sessions and off it working on geometrical proofs. We believe that this is certainly the most challenging topic in the A. Math syllabus. After working on so many questions, we find that to obtain the correct proof or solution, determination and some intuition is required. Even for O level questions on this topic, we find that very often, we’ll have to spend a considerable amount of time in order to arrive at the solution. Putting ourselves in students’ shoes, we feel that it will be very tough on them to require them to solve such questions under the constraint of time during exams.

This is not the typical A.Math topic which can be mastered easily by drill and practice. From feedback gathered from some students in my previous school, their teachers actually left this topic to the last and told them to concentrate on other topics during exams. It is thus easy to imagine the lack of confidence even teachers face with regard to this topic.

To help students, teachers have to be really proficient and not simply brush the topic aside. Once the confidence on our part is acquired, we can help students by working with them through numerous examples of various difficulties. Allowing for group discussion is also very important as students can question one another on the claims made. For tougher questions, hints should be provided so as not to scare students into “surrendering”.

Group F

Group A: Lab reflection

We had a 'compact' time through the geometry lab. Firstly some of us were new to the GSP software. While we learned to familarise ourselves with the controls and uses in GSP, we also had to focus on the geometry questions at hand so sometimes we managed to solve the maths problem but forgot how to use GSP or 'remembered' to draw in GSP but did not finished the question before it's time to move on to the next question. But i believe this will improve once we get the hang of it and we look forward to learning 'new things' that we can do with GSP.

It is good that GSP is introduced to us because it is another useful tool for teaching maths and the 'schools' uses it. So, is there some where we can practise or revise on GSP after the geometry lessons are over? Or are there similar free programs that is compatible to port GSP files over for use? I think it would be helpful if we can have the software installed into our laptop so that we can still practise in our own time.

Also it is interesting to know about the conventions for certain maths topics and how they practise it in schools. For example, for the 'Similarity' topic the convention is to state the reason in the final statement, 'triange ABC is similar to triange EFG (ASA)'. The practise in some schools state the reason ASA but some do not. And the way such questions are marked in prelim exams are different from the 'O' levels.

Geometry Problems

This week, we have explored the geometric properties related to triangles and circles.

In the last session, we have worked on four questions. The solution has been compiled and you can get a copy from http://sps.nus.edu.sg/~wujiangw/geometry.pdf



Group C

Assignment killer question (1) is on Heron's Formula

Hey guys, I found the following website while doing my maths reflection just now. The geometry question 1 that we are given in the assignment is actually a proof for Heron's Formula:

Preview of the webpage that constructed the proof: (diagram differs from assignment in labelling though)

http://jwilson.coe.uga.edu/EMT669/Student.Folders/Jones.June/heron/heronpaper.html

Cheers again,
Euclid

Trivial Proof of the formula used to find triangle in coordinate geometry (which can be extended to n-sided polygons)

Let say we want to find the area of the pink triangle (A+B) given the coordinates of the three vertices

Similarly, we can prove by the box method and construct a different diagram to obtain the area a:

Cheers, 

Euclid

P.S sorry for the messy print screens. The equations cant show up properly in the text box.

Teaching Site

A hyperlink to the Teaching site is added at the  corner :)

Courtesy of Group E :D

TLS: Angle at center is twice angle at circumference

We have learnt that the above approach can be used to show the angle property of angle at center is twice angle at circumference.

Thanks to Junyang and Sharon's question, how will we facilitate the discussion for the explanation/proof of this property if it is of the case below? 



Hint: think about angle in semicircle. Some of you may think that this is circular argument as angle in semicircle is derived from this angle property. However angle in semicircle can also be explained by two isosceles triangles.

Group D: Ivan Sutherland

Hi All,

Did you know that the sketchpad was created by Ivan Sutherland?

Watch the videos below to learn more about Sketchpad.

http://www.youtube.com/watch?v=USyoT_Ha_bA
http://www.youtube.com/watch?v=BKM3CmRqK2o

Hope you will use it more in your lessons next time.

Group D

Reflection by Group D:
One thing we learnt in today's tutorial was to use the properties of congruent triangles.
Triangles are said to congruent if they have the same shape and same size.

Congruent Triangles
There are four properties of congruency in triangles.

1. The SAS Property:
   
   If two sides and the included angle of a triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If AB = DE, BC = EF and the included angle BAC = the included angle EDF.
     If the above conditions are satisfied, then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Side, Angle, Side property.
   
   
2. The ASA Property:
   
   If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If BC = EF, ÐB = ÐE and ÐC = ÐF.
     If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Angle, Side, Angle property.
3. The SSS Property:
   
   If three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If AB = DE, BC = EF AND AC = DF.
     If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Side, Side, Side property.
4. The RHS Property:
   
   If the hypotenuse and a side of a right-angled triangle are equal to the hypotenuse and a side of another right-angled triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     ÐB = ÐE = 90°, side BC = Side EF and hypotenuse AC = hypotenuse DF. If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called the Right Angle, Hypotenuse, Side property.

Reference:
http://www.kwiznet.com/p/takeQuiz.php?ChapterID=2817&CurriculumID=24

Geometer's Sketchpad

I am surprised that none of my teachers have taught me using Sketchpad before. Also many of the classmates in QCM520 have not used Sketchpad or been taught Sketchpad before.

According to my experience, i have not seen any in service teacher use sketchpad to teach before even though the Sketchpad is very useful.

I wonder if all schools have access to this tool or if there are other reasons why schools are using sketchpad more.

sketchpad documents can be exported to websites for posting

GROUP D - STEPHEN, HAFIZAH, DANIEL, JUNYANG

Reflection on Expostional Teaching

Last week, we have done expositional teaching in the tutorial sessions. Here are some of things that we have learnt from this exercise.

First, it is on time management. It is a natural tendency for us to try our best to teach as much as we can in the class. However, one thing that we have overlooked is how much the students can learn within one period. We are fortunate that we do not have classroom management issues last week. Once we stepped into a real classroom, we might need to spend some time on it thus we need to think of how much time we have and how much we can cover. Our group tried to cover four factorisation methods within forty minutes using quite a number of different examples. However, the time devoted to each method for the students to learn is limited. It would be more effective if we had focused on one method and give more time for the students to learn and apply.

Second, it is on the choice of instructional method. Our group made use of both powerpoint presentation and whiteboard for teaching. During the planning stage, we are not aware of the size of the whiteboard. On the day itself, we found the space available is quite small thus many workings and steps are not presented in the way that we have planned to do so. Only when group E was presenting, we realized that we could make use of the visualizer instead, because it would be independent of the room that we use. As learnt from yesterday's lecture, we could have checked the room in advance and plan the lesson based on the room that we use.

Since we are learning through this exercise, I think it would be helpful if other groups can provide us with some feedback on our teaching. Please feel free to talk to us in private or just leave a post here.

Group C
(topic: factorisation)

Reflections on Expositional Teaching

Realised through the expositional teachining review that teaching of the right concepts is of paramount importance in the teaching of mathematics. While creativity is welcomed in the presenting of concepts to our students, it should be properly thought through to ensure that teachers not do it at the expense of contradicting basic concepts.

Even last week's lecture has emphasised the point that even if one is using direct instruction to teach, the key thing is to teach the 'right thing', the key concepts, the fundamentals must be communicated through to the class so that there would not be any misconcepts that students need to unlearn as they progress through the lesson.

Grp D

Some thoughts on expo-teaching

Ref to lesson on simultaneous equations

Just felt that it would help to point out the number of equations needed to solve for certain number of unknowns. For example:

2 unknowns x,y will require 2 or more different equations involving x,y so as to solve.
goes on for 3, 4...

Might help the students to pick out enough key points in word problems so as to form the required equations.


Hope it helps
Group B

Use of teaching aids in class

Last week, we have looked at the use of teaching aids in algebra. We first started with alge disc and alge tiles in term of expansion and factorisation and then four pan balance for solving linear equations. These tools do help us to bring in the concepts of operations thus let the students understand the underlining meaning of each step we do. However, once reached the negative values or fractions, it may be tough to use these tools. Actually I felt that we should not force the students to use the teaching aids to solve the complex questions but rather let them use the embedded concepts to solve them. By doing so, we can get them smoothly transit to abstract thinking which is very important in upper secondary and pre-u level as some of the topics cannot be taught using models. For example, for quadratic manipulations we can use a 2d model and for cubic manipulations, we can use 3d model but it is more difficult. However, when we start to talk about higher powers or fraction powers, we cannot really use them any more. Thus it is important to use the models for conceptual learning and also ensures that the students think about what they are doing as abstract mathematical manipulations at the same time.

Group A

We have learnt in class that a 4-pan balance can be useful in helping students understand solutions of linear equations better. We had fun playing with the balance in class as well.

I would like to share some sites that contains animation of pan balance to help students build up their algebraic thinking.

Pan Balance – Shapes 

One of the more interesting site, it allows students to guess the various weights of the shapes while incorporating the fun factor.

Pan Balance – Expressions

Expressions illustrate the relationship between 2 equations. However, it would be more useful for students who are learning simultaneous equations or solutions of equations using graphical method.

Pan Balance – Numbers

Numbers is useful to let students have a feel of what an equation means. It emphasize that equality is a relationship, not an operation.

The sites are self explanatory with clear instructions within it. Have fun exploring!

completing the square by using algetiles

Reflection (Group A)

The tutorial lessons are varied and interesting unlike our 'student' days when we dragged ourselves to Math classes. As we go through each lesson, we can see that there are different materials to help us introduce Maths topics to the students.

In Ratio and Proportion topic, for example we can use newspaper articles showing football penalty statistics to set questions for students to apply ratio. We can use YouTube video to show them the famous Golden Ratio and task them to measure the span of their hand against the forearm to see who is the 'golden' child.

In Algebra, we can use alge-disc, alge-tile to show the operation and factorisation of Algebra expression. And use Beam-balance to show the 'balancing' in Algebra equations.

Going through these exercises brings 'life' into the subject and make us think of how other resources can be incorporated into our classroom lesson to make it fun and meaningful.    

Reflection on algetiles

   We further make use of algedisc to explore more difficult questions and used algetiles as another form of tool to algebra questions. Algetiles give a more geometrical picture which our group prefers. These tools help the student to visualize equivalent expression and will benefit students who are better visual learners.

    When using the algetiles to try to solve x² – 2x, we discussed using the ‘1’ tile to cover the protruding portion. In trying to explain in this manner, would it mislead students to think that the ‘x’ tile is equal to 3 ‘1’ tiles since we deduce that the protruding portion is equal to ‘1’ tile and 3 ‘1’ tiles will fit to the size of the ‘x’ tile? From our group, there has been experience that students have this wrong idea where ‘x’ equals to 3 due to the tile size proportion. We should take note of this possible misconception among students when using the algetiles.

Cheers,
Cheah Boon (group F)

Fun with Alge Tiles

      complete the square?

Okay. So we were having fun with alge tiles and in a way we do appreciate how it can be a useful tool. Then Mr Tan was sharing that while algediscs might be readily available, we would need to custom make alge tiles ourselves if we would like to use them.

So while we were measuring the dimension of the tiles that were used in class (its 1 by 1cm, 1 by 10cm and 10 by 10cm by the way), we came across an idea to improvise on the design.

      visual aid for visual people

Instead of using Red for the positive terms and Blue for the negative terms, we can try keeping Red for the positive terms but white for the negative terms instead. This way, when we cover the red tiles with the white tile, it would look like that the portion of the tile is gone.

And to enhance the effect, we can leave out the labels '-x, -1...' 

Anyway there are many designs of alge tiles out there and here's a 3D one...

In any case, alge tiles can be fun and when used appropriately can help to, as wikipedia puts it, allow students to better understand ways of algebraic thinking and the concepts of algebra.

Here's some creative art work by students using alge tiles. (but they paid attention and did all the questions in class)

Group 5

Flash mind reader - Charanya

Hi guys,

After today's tutorial I came back to look for that "computer reads your mind" game that Daniel showed us today. I came across this - so just to share this online version of the game for those who are interested

http://kids.niehs.nih.gov/mindread/mindread.html

It is called Flash mind reader, the explanation is also given in the website.

Charanya

Reflection by Descartes

The tutorials have been well structured to induce thinking and understanding. Mr Tan, our tutor, rigorous reiterations have provided a deeper insight into concepts otherwise taken at surface level. Various avenues to tackle a problem were addressed and discussed. All these add on to a greater appreciation for the works we will be teaching and also cater to a meaningful learning experience.

Having said that, the meaningful learning experience seems to be what we are gaining and time will not be on our side for it to be transferred meaningfully. It cannot be denied that such in depth discussions will be done at the expense of completing the syllabus. Of course, students will achieve a holistic meaning to what they will be learning but how much can they apply? Examinations have become almost predictable. Most students are comfortable with practice makes perfect. Doing sums from ten year series or whatever practice books they can lay their hands on over and over again will over time help them to recognize what a problem wants without knowing what the underlying meaning to the problem. Just like some Chinese students from China who speak little or no English can recognize the process of getting the answer to a problem just by identifying the key words in the question. So to say, rote learning does have its merits. Afterall, most of us were trained for the exams.

The problem is all about igniting the learning process of the people of today’s generation.

Back on to the lessons reviews, it had been an interesting one with the algebraic discs. Algebraic discs used in class are great hands on approach. But it can only serve that much. The principles in using it contradict how we were taught. So much in the sense that we ourselves are forced to conform to another method of learning altogether.

The ratio and proportion activity was a good startup especially in garnering students’ interests in the topic itself. The discussions brought along by it showed how similar problems can be represented differently. Fibonacci Theorem and the relation to the golden ratio were further explored with a very engaging video.

Friday’s lesson on percentages and indices was like playing mind games with us. But of course the takeaways are that diagnostic test for topic on percentages proves to be a great introduction for lessons to get students thinking and for indices, its better not to have too complicated thoughts. Definition can only be verified not proved.

Anyway, there are various math forums online that reach out to the bigger community. Just try googling. Not forgetting Youtube has various proving of many formulas such as in the topic of mensuration.Yes, today’s IT age has made information so readily available that teachers have to keep up with the times and constantly think out of the box to sustain the attention of the Y-generation. This is definitely a good platform for relevant discussions and resources to be compiled. Thank You for the initiative and we hope to make this blog a success!

TLS: Balance between teaching for understanding and efficiency for examinations

Just to reiterate the idea I was trying to bring across today in response to a couple of posts. I'm on the same page as you all on the constraints in the real classrooms. It's important to try and balance both teaching approaches in our classrooms. Let our students at least experience once the meaning making or sense making of certain topics. After that we should have all the creative ways to let our students learn the various procedure in solving Math problems and examination type of questions efficiently.

BTW, some of you are asking about how to access the resources in the tutorials. You can access them here, bear with me as it's still work in progress.

My reflection...

I must say for sure that the past few tutorials had really broaden my view on Mathematics in particular to the topic on Arithmetic. Thoughts have been constantly provoked throughout the lesson which I felt that it was a better way of learning as compared to a one-way tutorial session (i.e. tutor keeps talking with minimal interactions among the class).

By crafting lesson using the TfU package (teaching for understanding) will definitely benefit students in a long run.   But for sure, we will not be able to design TfU package for almost all the topics due to time constraint. So in this case, how should we decide what topics to be covered under the TfU package? Should it be based on how important the topic is or should it be based on how interesting the topic will be for the students?

As in the case of the algebra discs, I am just wondering whether students in future will be overly reliant over it? Given a question e.g. [-3(-2)+4]/3, will they be able to solve it immediately? Or will they spend time breaking up the expression into separate terms so as to try to fit into algebra discs. By the way, I found this journal paper online "ALGEBRA DISCS: DIGITAL MANIPULATIVES FOR LEARNING ALGEBRA" written by MOE Algebra Team which may be useful to some of you. Below is the site for downloading:
http://tsg.icme11.org/document/get/264 


Jacqueline

Cheng Xiang's Reflection

Through these 2 weeks of lessons, it is rather clear that the emphasis of the course is on having teachers adopt a “teach for understanding” stance.  While I agree that it is important for concepts to be thoroughly understood to allow for a meaningful learning experience, I feel that it could actually bring about more problems in the actual neighbourhood school classroom setting.

Many students are practical learners. Those who want to learn have the aim of doing well for exams. As such, they might lose interest if teachers dwell too much on the understanding of concepts like (-1) x (-1) = 1 and the rationale behind the ladder method for LCM. I can already foresee students asking questions like, ”Cher, textbook says negative times negative gives positive, I just follow and I can solve the questions. Why do you need to explain in such detail?” and “The ladder method I know how to use. But the rationale behind it, do I have to know?” These students could end up getting bored and might even lose interest in the subject, simply because such a degree understanding is not required in the syllabus. Unless the syllabus is altered and exams are geared towards testing for understanding of concepts to such depth, students will likely not learn to appreciate. Moreover, such explanation of concepts in depth might be a tad challenging and time-consuming. This might not be practical in the all-so-familiar scenario in which the syllabus has to be completed on time.

That said, I believe that such kind of “teaching for understanding” approach that allows for in-depth discussion of mathematical concepts are more suited for students in the Integrated Programme. These students bypass the O levels and have more time to engage in meaningful learning. They could thus be more receptive and in my opinion, should be the true beneficiaries of this teaching style. 

Cheng Xiang

Self Reflection on Maths tutorials

For the past 3 tutorials, I had been learning new things after every Maths tutorial. Mr Tan had been throwing many questions that were quite provoking and challenging to my thinking towards to what I was taught in school as a student. It kept me wondering that as a student, I accepted what was being told and didn't ask any questions to my teachers. Unlike here in the class, Mr Tan had been bombarding questions, ideas and suggestions so that it kept the whole class think and don't accept anything that were told in class.

I learned that emphasized on the concepts is more important rather than getting the final answer. Simple yet important key concepts in real numbers, place value, addition, subtracting, multiplying and dividing of negative numbers and HCF and LCM. I agree with the rationale of going down to the level same as the students so as to understand what difficulties they face when the teachers teach maths in class so that we can better explain and ensure that the students will able to have a meaningful learning.

I agree with the notion of infusing meaning and use real world context into lessons. For example, we did spend an amount of time trying to make sense of how we can teach the operations with negative numbers. It is good if we are in ideal situation. In reality, there are many constraints that the teacher is facing and sometimes, it is near impossible to use this kind of approach for all topics. However, teachers need to find ways to ensure some meanings are being taught to ensure clarity and understanding in lesson.

Firdaus

Reflection on past three tutorials

In the past three tutorials, we have looked at the mathematical problem solving and teaching of arithmetics.

In the first session, through solving a few mathematical problems, we have revisited the pentagon model for solving mathematical problem. It is important to note that the five elements are required for the problem solving but at the same time, the mathematical problem also reinforces the five elements.

In the subsequent two session, we have looked at the skills for solving some arithmetic problems. However the emphasis is not on the algorithm but the underlying concepts involved. Through these few exercises, i learnt how to apply the concept of addition, subtraction, multiplication and division in explaining the mechanical operation involved. By using the negative numbers in the examples, it helps me to re-think what I knew in the past and thus understand the concept better.

At the same time, I also realized that it is important to bring us down to the level of the students and use more concrete or pictorial examples rather than abstract concept in teaching to help to communicate to the students.

Reflection by Alex

Hi all,

My personal reflection is that I have learnt a lot in improving my Math concepts which we didn't get to master even as a university graduate.

In general, for the past few tutorials, IMO, the questions raised in the tutorials in teaching were covered and discussed but some methods of getting the answer and answers were flashed over too quickly for us to copy down.

An example would be Find last 3 digits 1995^1995. It would be good if the answers to these questions are also available after they have been covered online or otherwise.

IMHO, for today's tutorial, too much time was spent in explaining the multiplication of negative numbers. It would be easier and faster to explain division and then multiplication of negative numbers using the same concept instead of multiplication then division so that the remaining questions are not rushed over.

The tutorials can also be made available earlier after the lectures so we can have time to try the questions before the tutorial and more specific deadlines and expectations for assignments and submission should be announced.

Despite this, I can sense your passion in imparting the teaching of concepts instead of just technical skills and formulae and also salute your desire to improve our tutorial. Hope you will keep it up!! =)

Aili's reflection on meaningful maths lessons

The previous two tutorials on arithmetic have been pretty mind boggling for me. In order to infuse meaning into our lessons, we are supposed to explore the possibilities of using real world contexts to teach mathematical concepts and try to make sense of these concepts to our students. As a student-teacher, I firmly believe that this shift in pedagogy is good for our students as we need to move pass traditional rote learning through memorization and repetition.

However, there are several difficulties and limitations to achieving this especially in a realistic classroom setting with syllabus to complete within tight schedule and preparation for exams on the students’ part as well as disruptions from difficult students. We have already experienced the potential disruption to the pace of teaching something fundamental yet abstract (multiplication of 2 minuses make a plus) in an attempt to make sense of it during tutorial. I have yet to find a satisfactory meaning to that concept personally but there are several logical explanations and examples available to help teach this concept (http://www.mathsisfun.com/multiplying-negatives.html). Furthermore, there are also concepts such as higher calculus (differentiation of tan x) which can be hard to infuse meaning when teaching. (We can try to facilitate learning by showing the derivative of tan x is sec^2 x with graphical method to trace the gradient => back to concrete-pictorial-abstract pedagogy instead of pure memorization of the derivatives).  We also cannot neglect the value of rote learning altogether as I feel that practicing the processes involved in problem solving can build up familiarity and confidence in addition to skills.

In conclusion, the onus is really on us to decide for ourselves the delicate balance of what to meaningfully teach our kids and what to drill them on.

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