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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