Teaching algebra

It is my second year teaching P6 Standard Maths and second year teaching algebra. Compared to last year, I feel that I am more explicit and clear in my teaching. I am mindful of the use of terms and always try to make the meaning of the terms explicit. For example,

(1) Meaning of the variable : Represent unknown number

(2) 3x : Instead of saying 3x, I always says 3 groups of x

(3) 3+x : You cannot add a know number to an unknown number

(4) Use of model to represent algebraic expression

Even getting the students’ response, I also make sure that they also articulate the meaning of the terms.  Unlike last year,  I did not try to teach them “mechanically” but rather trying to get them to articulate the meaning.

How is that going? So far, they seem to catch on (at least during the class activities). Let’ see how that translate to their actual test. I am simply keeping my fingers crossed.

 

 

 

Gap, Gaps and more Gaps!

As I reflect on what happens in the first two weeks of the school, I realize that I am more aware of students’ responses and try to build on them. I am trying to address the students’ gaps  based on their responses. Here are the responses that I have received so far and how that influence my next step :

(1) Students’ not too good at mathematics fluency as articulated in the previous blog post [Link]

(2) Students’ alternative concept with 3+y and 3y (From the Nearpod)

(3) Not able to recall the photosythesis (Through journal entries in biology textbook and MCQ in Activity book)

(4) Not able to use “electrifying” terms (Through journal entries in biology). Observations from the journal entry:

– Just simply using conductors and insulators. Do not talk about the current flow in the description.

– Talk about the topic generally, without applying to the context.

One of the kneejack reaction to all this is to teach more!! Just tell the students and you expect them to remember. I guess I am not going to do so much of that but rather design learning activities to get them the make their thinking visible.  These are the activities that I have planned for Week 3:

(1) Maths Journal Entry : Get them to articulate the “why”  in the “commonly wrong” fundamental maths problems. This include journal prompts like Which number is bigger : 1.6 or 1.09 ? Explain your answer

(2) The problem with 3x and 3+x . Will likely to get them to screen cast the working and get them to talk about it.

(3) Science Blogging : Getting them to write a letter describing what they have learned about electricity for the past two weeks. As a follow up activity, I will get them to peer critique the students’ posting withe the success criteria provided.

(4) Get the kids to create a mnemonic to remember the facts.

 

Mathematics fluency (a 10 minute exercise that reveals my students’ learning gap)

For the past one week, my students would have mathematics fluency exercise at the start of the lessons. I have tried to vary the exercises, from using (1) two dices for them to form number statement (using the animation at : http://a.teall.info/dice/ )   (2) Socratic App (3) What makes 8?

This simple 10-minute exercise reveal the students’ learning gaps in fundamental mathematics fluency. This includes

(1) Not able to tell which number is bigger : 1.09 and 1.9

(2) Not able to covert improper fraction to fraction

(3) Having difficulty in identifying the digit in the tenth, hundredth place.

(4) Not fluent in multiplication facts.

Well, this shows that there is lots to be done to close the learning gaps. I will try to cover the fundamentals (e.g., place values before I go on to more complex stuff).

Am I daunted ? No! My last year’s experience with my beloved  rascals last year  has made me feel “zen” with such learning gap.  I will enthuse this group of students like what I did last year. I will pull no stops to make them step up before the upcoming PSLE!

 

 

Teaching P6 Std Maths for the 2nd year

It is my second year in P6 Maths. Here are things that I will need to do to design meaningful learning experience for the children. One key point that I must always remind myself is to make my thinking as visible as I can.  Things that come naturally to me needs to be explained as these things might not be so natural for the students.

There would be structures in the lessons so that students would be familiar with the routine.

Structure of lesson based on Gradual Release of Responsibility 

10 min : Factual Fluency + Review of the previous lesson

15 min  : Focused instructions : Learning new concept (Activities to introduce them to new concept)

20 min :  Guided Instruction/We do it together : Doing textbook/workbook exercise

5 min :   I do it on my own : One question to solicit their understanding on the new concept (Exit Ticket)

5 min : Buffer

 

Things to do in the lessons

(1) State the learning objectives for the week/day. Be explicit about that.

(2) Have a regular bite-size quiz for the students (twice a week)

(3) Give them general steps in solving problem. For a start, I would be doing geometry with them. Here are the steps to guide them

(a) Label the diagram with all the information you have gathered from the question (including given angles, sides, parallel sides)

(b) Label other known angles  using the following property:

– Angles opposite the equal sides of an isosceles triangle

– Vertically opposite angles

–  Angles opposite in the parallelogram 

– Angles in the equilateral triangle, square, rectangle 

(c) Start from the angle that you want to find.  Determine how you want to find the angles using the following property

Other Angles

– Angles on a straight line

– Angles at a point

– Angles between parallel lines

– Sum of angles in a triangle

Same Angles

– Angles opposite the equal sides of an isosceles triangle

– Vertically opposite angles

–  Angles opposite in the parallelogram 

– Angles in the equilateral triangle, square, rectangle 

 

Making afternoon 2-hr extra lesson count!

It is the PSLE season. So, what is the No 1 to-do to help your students? *Drum roll* It is none other than extra lessons!!

Yes, l hold extra lessons for my Maths class too.  That is important for them as at least they do some meaningful watch under my watchful “hawk” eyes.  I am trying to fine tuning my extra lessons so to keep them focused.  After all, getting students to study in the afternoon for 2 hours is no easy feat.

Last week, I allowed them to do a Paper 1 within 50 minutes and went through the paper with them. I personally felt that the session could be better as I can sense my students were only physically in class.  They were losing focused and yes, they were getting bored.  On reflection, this could be due to the lengthy going-through (about 1 hr) after doing the paper.

For this week lesson, I adopt a different approach. I break the lessons into a series of 4 1/2 mini lessons.  For the first 2 lessons, it is about revising volume and algebra concepts.  Within the half an hour, they are to do some questions with their peers. Then, I will round up the lessons by going through with them.

For the final 2 lessons, they are to do Paper 2 (about 5 questions). They are to spend about 30 minutes on the the 5 question  For the first 15 minutes, they are to do it on their own before discussing with their friends. For the final mini lesson, I invited them to share their answers on the board.

Overall, I feel that my Week 2 extra lesson is more fruitful. This could be due to peer learning and the mini-lessons. They enjoy learning with one another even though they might get rowdy at times. There is variety in the lessons and the kids do not get bored. Also, the questions and the concepts chosen are “reachable” for them.  It is not too easy but not too difficult for them.  It makes me happy to hear them thinking aloud and pointing out their peer’s mistake.  

I have to admit my boys are unfocused at times and need my “not-so-friendly” remainders constantly. But, what is new? Boys , being boys, will always have the knack of making teachers angry :-).

Strong Fundamentals

Have some in-dept discussion with my colleagues on Mathematics learning and here is my reflection on the discussion.

In Mathematics, fundamentals are very important. If such fundamentals are not built up in their younger years (P1 – P2), the problems would surface when they grow older. Without fundamentals, they might not have number sense.  The numbers do not “talk” to them and perhaps they are just symbols without real meaning to them. Such weak fundamentals would cause the kids to have difficulty in solving the more difficult word problems.

Sometimes, I wonder what causes the weak fundamentals.  Is it the lack of emphasis on fundamentals in lower primary school years? There might be some case of “over-testing” and thinking more difficult exam/test papers would help the students. The over testing may mean teachers over focus on age inappropriate word problems Does it really help? My personal opinion, it is a big “No-No”. I believe in building strong fundamentals in the lower primary. If the students can understand and master the fundamental concepts, that would help them when they grow older.  Put it simply, if the student truly have a deep understanding of :

  1. Part-Whole model
  2. Comparison Model
  3. Fraction of A Set and Fraction of a whole

They can solve almost every problems.  The students only need to extend their schema (based on strong fundamentals) to solve the more difficult problems.

At upper primary  level, the problems get tougher and more complicated. For students with weak fundamentals, there is no way they can extend their schema. They are perhaps building new schemas and this may prove cognitively challenging for them.  They can’t see the connectedness with the different concepts.

So, what can we do about this?  Do we have time to go back and rebuild the fundamentals? Dow we have time as we need to “cover the syallabus”? That is the struggle and tension that I am constantly facing.  And I do not have an easy answer for that.

 

Mathematics

Author : J R

CC by 2.0

 

 
Continue reading “Strong Fundamentals”

There is more to learn from wrong answers

Teachers generally enjoy seeing the “right” answers from the students. So, what about “wrong” answers? Wrong answers also can provide learnable moments as we get the students to do the error analysis. We should refrain from giving them corrections and make it a penmanship exercise (“Students, please copy down the answer”) Rather, we can allow the peers to fix the errors.  but rather  we fix it for them. Afterall, we seek to make our students work harder. Fixing the errors is one of the formative assessment strategy and through such activity, we are activating the students as instructional resources.   Read more on why the teacher need to know answers at this link

 

For the past 2 weeks, I have been doing that with my Maths class. I feel that they have attained the level and should be able to identify the errors.  I allowed them to comment on the solutions or ask them to interrogate the pupils. Here are samples of their comments.

Fix Error 1

 

Fix Error 2

 

This is a fruitful exercise as I am getting the kids to think. And they do not disappoint me and show me that they can pinpoint the mistakes. The challenge for me is to find “fixable” errors for them and pitched at the right level. I also hope this exercise can also boast their confidence for mathematics.

 

 

 

 

Making Seatwork Visible to all with technology

Doing seatwork is a regular feature of my Maths Class.  Sometimes, the kids might not even do it as they are either not interested or feel that the work is beyond them.  For this week, I decide to give such seatwork a twist. I am going to make the seatwork visible to all by use of technology. It is still doing the same seatwork but their seatwork would be visible to all so that others can critique on it. Essentially, there is audience for their seatwork and it is not just meant for teachers’ eyes.  Here is how I conduct the lessons:

  1.  Getting the students to work in pairs

There are two reasons for this. Firstly, I want the students to give each other support (activating peers as instructional resources). Secondly, this is to “half” the possible technical issues arises. I have tried getting them to use 1-2-1 iPads but the logistics is a petty nightmare. (I will post about on to how to manage the iPads in one of my future posting)

2. Use Nearpod so that students can ink the answers. The answers would be then shown on the projector screen almost immediately after they have submitted answers. Since there are only 7 groups, it is petty easy for me to have a quick glance at the answers and make a quick assessment of their learning. I will then get them to discuss the answers given and how we can improve the answer.  The best answer would be send to each individual iPad for viewing. Here is a sample of the working,

 

Students are doing such seatwork and they “insist” that they have to do this more often. If they discover their answer is wrong after reviewing their peer’s answers, they would want to modify and resubmit the answer. They can also point out the mistakes of their peers’ answer (like using the wrong radius or omission of units).  The reward for the best answer would be sent to each iPad for viewing. Some students really aim for such reward and try to give me the best answer in the shortest time. Their “super-on” reaction to such small award make me realize that my students needs affirmation. Guess I must nag/scold less and praise more.

It is indeed gratifying to see my students doing such seatwork enthusiastically. It gives me the much needed energy boast to do even more for them (even it means less-me time). Sometimes, teaching them can be very demoralizing when you realize how wide their learning gap is. Even the teacher (aka me) also need some affirmation from the students that we are moving in the right direction.

 

 

 

 

 

 

 

The “Fixed method” is really fixing up my students!

Thinking about the question. That is one of the weakest point for my Maths students. They tend to want to just do the “fixed method” . In problem on quadrants and semi circle, they decided to add in the diameter whenever they see the semi circle and quadrant. Even though they have correctly outlined the figure, they do not seem to relate the outline to the perimeter.

It is really challenging to get them to stop and think. After all, I am fighting the habit that they had for the past few years. Why do they have such habit? Have they been schooled into remembering the “fixed method” since they can’t do maths? Or is that the only strategy they have when solving maths problems?

I am not giving up and would continue to press on. I admit that it could be frustrating at times but I have to keep my frustration in check. It is no point “thinking they ought to know”. Since they are in my class now, it is my duty to make them learn. I can’t promise I will succeed but I am definitely going to do what I can .

All about circles – Making learning concrete!

Currently, I am teaching my class circles. To make their experience concrete, I have done the following :

  1. Get them to draw the circles and cut out the circle. Emphasize that the outline is the circumference
  2. Get them to fold circles so as to know about quadrants and semi circle
  3. Getting them to experience fraction disk. I “morph” the fraction disks and teach them about circumference.

 

So, far , the students seems to be getting the hang of it except for a few. Hope that the next few lessons would help them to reinforce the concept. They still have problem remembering the formula.