As all mathematics teachers know, “before/after” problems are classic problems that our students will see most of the time.
Here are some of the tips that I have tried to teach my students.
(1) How is the “before bar” and “after bar” related? That is the most difficult part. Once the students can see the relationship, half the battle is done.
(2) Is there any implied information given in the problem ? For example, I try to get my students to see if this is a special case of before/after. Like “You give me, I give you” (where the total is constant) and “Both increased or decreased by the same amount (implied difference is constant)
(3) Is the “before bar” and “after bar” the same ? Any information given in the question that implied this
This is a very broad principles that I have when I am teaching the students the “before/after”. Of course, when I am teaching that in class, I need to unpack the questions to even more minute questions so that the students can handle. This is a skill that I am still refining. Feel free to comment if you have more tips.
Ahmad and Davis shared a box of sweets equally. Ahmad ate 15 sweets and Davis ate 23 sweets. After that, Ahmad had twice as many sweets as Davis. How many sweets were there in the packet at first?
Another way to solve this before/after problem with MaPS. Hope that this will give you some ideas on how to “tackle” such classic problem.
Ahmad and Davis shared a box of sweets equally. Possible questions to ask : What does “equally” mean? Do they have the same amount?
Ahmad ate 15 sweets and Davis ate 23 sweets
Again break this statement into two part. Let’s take a look at the first part “Ahmad ate 15 sweets”.
Possible questions to ask: How do I show “ate 15”? Do I take away from the “before bar” or do I add on to the “before bar”?
Ahmad ate 15 sweets and Davis ate 23 sweets
Now for the second part” “David ate 23 sweets”.
Possible questions to ask: How do I show “ate 23”? Do I take away from the “before bar” or do I add on to the “before bar”?
After that, Ahmad had twice as many sweets as Davis.
This is the tricky part. Guess we must be go slower at this point
Possible questions to ask : Who has more now ? How do I show “twice as much”?
The model is now almost completed. Hopefully, the students can see the “answer”. If not, we will need to prompt them with questions like “What is the value of the yellow part?”
John had $40 more than Ali. After John spent half of his money, Ali had $20 more than John. How much money does John have at first?
Here is a typical before/after word problem.Again, I shall tackle this word problem “statement by statement”.
John had $40 more than Ali.
This is a simple enough statement that most pupils can draw (I hope :-)) I will be showing the screenshot of the model (instead of video) Possible question to ask : Who has more ? How much more?
After John spent half of his money, Ali had $20 more than John
Break this statement into 2 part. Let’s focus on “Ali had $20 more than John”. Possible questions to ask: Who has more now? How much more? How are the “before and after” bars of Ali related ? Did Ali spend any money?
After John spent half of his money, Ali had $20 more than John
Now, let’s handle the first part of the statement (“After John spent half of his money”).
Possible questions to ask: Does John has less money after spending money ? How are the “before and after” bars of John related related ?
The final step is to draw the dotted lines to make the model more clearer. Possible questions to ask : How are the “20” and “40” related to any parts on the model?
Kai Yee, Lalitha and Monica share a total of $963 among themselves. If Kai Yee has twice as much money as Lalitha; and Monica has $18 more than Kai Yee, how much money does Kai Yee have?
In a step-by-step approach, we are going to unpack each statement into the models. It is okay that we do not know the exact size of the bars at first. As we go along. we have to resize the bars. The models might not be accurate at first and it is okay to resize the models. This may help the weaker students. This point is brought up in the article at the Mathematics Educator
Kai Yee, Lalitha and Monica share a total of $963 among themselves. Possible Questions to ask : How many bars must we draw? How do I show $963 ?
If Kai Yee has twice as much money as Lalitha Possible Questions to ask : Who has more , Kai Yee or Lalitha? How do I show “twice a much”? Who has 1 part? Who has 2 parts?
Monica has $18 more than Kai Yee, Possible Questions to ask : Who has more, Monica or Kai Yee? How much more?
At this point, it seems that all the information has been presented in the Model. We have to prompt the students to look out for the “implied” information.
Possible questions to ask :”Can we solve the problem by just looking at the model? Is there any information that we can put in ? How are the bars of Monica and Lalitha related?”
That is all for now. I will leave it to you to solve the rest of the problem. Too lazy to continue:-) Hope you find this useful!
Most of us are proficient Mathematics problem solvers such that we are able to “see” the solution once we see the problem. Even in drawing the model, we have already “see” in our head how the final model looks like. However, this is not true for our students. They are still novice problem solvers and cannot “see” the solution. Hence, it is important to make our thinking visible and unpack our thinking process into details such that our students can follow our thought process. I tried this with my class last year and it is not a very straightforward process. I have to really make a conscious efforts to break my thinking into steps that my students can follow.
In this series of word problem solving posting, I try to model such approach. Please note that I am doing this in my free time, so do not expect perfect work :-). This is in no way endorsed by the authorities but purely just me and my crazy idea of sharing my 2 cents worth on problem solving. Hope that the fellow Mathematics educators will find them useful. My approach of solving word problem is based on the following principles:
(1) Systematic and explicit instruction (Step-by-step approach)
(2) Self-instruction/questioning (For monitoring and management of their learning)
(3) Use of visual representation (Drawing of Model)
As Mathematics teachers in Singapore, we have always heard about CPA (Concrete – Pictorial-Abstract) approach in teaching of mathematics. With the increasing popularity and the easily availability of virtual manipulative online, there is a need to include the manipulative approach. In the last sharing in TSG 18, Dr Lee shares about the CVPA approach (Concrete Virtual Pictorial Abstract). In his study, he has worked with some schools. Virtual manipulative can help to improve both the efficiency and effectiveness of the classroom instruction.
For more information, you can download his article from eJMT website
It is important for Mathematics teacher to ask questions in class. They should spent about 30% of the time answering questions. The questions should be able to provoke deeper understanding the lessons. Teachers should move from knowledge dispenser to knowledge constructor. It is important to think about the possible questions to be asked so as to bring about deeper learning in the students. When doing the analysis of the conversation in class, it is possible to consider the following coding schemes:
(1) Understanding Students’ Thinking
Telling Answer, Asked closed questions, request explanation, pressing for vigour Best Practices
Constructs knowledge from conjectures, Facilitates reasoning
(2) Yes / No, Name/ state, Describe/ explain questions (Hiebert et al, 1999)
“Belief about mathematics and problem solving (orientation) may influence the choices made by the solver” (Schoenfeld. 1985). It is not doubt that emotions do influence positive solving process. Surprisingly, a review of meta-analysis of non-logitutinal surveys and longitudinal reviews and PISA data reveals mixed results. It might be influenced due to age and culture.
Emotions are essential part of the problem solver’s regulation. It has been suggested that basic emotions (anger, sadness, fear, respect surprise and disgust) rarely occurs in basic problem solving process Emotions has physiological, psychological and social function in self-regulation process. Physiological adaptions makes the body ready to face the challenge. In psychological self-regulation, emotions biases cognitive process and direct attention.Emotions also play the adapting function in social coordination of the group. Negative emotions are part of productive process. Students must be taught to cope with such emotions by joking. It is important that teacher should acknowledge such emotions in problem solving and plan task that is manageable for the students. There is a need to teach teach the students explicitly how to handle such emotions. For example, 5eachers could model the appropriate enthusiasm and self-regulation behavior in problem solving process.
In East Asia, the mathematics education is more inclined towards product-oriented approach in which the students have to be grounded in their content knowledge. There is strong belief that the students must have “product” in which they can develop the process. In East Asia, teachers usually adopt didactic teaching in which students are passive receivers of information. East Asian countries are heavily influenced by CHC (Confucian Heritage Culture). The East Asians are more compliant, obedient and respect for superior. On the other hand, the western countries see the value of Mathematics in its process (of analyzing the reality). Thus, the style of teaching is more process-oriented and individualized learning. The students are seen as active construction of their knowledge and mathematical communications and reasoning.
In Japan and Korea, parents spend more than USD$10 billion on tutions. In Japan, it is called Juku schools while in Korea, is is known as tutorial school. Usually, such tution schools advocate the memorizing of the mathematical facts.
So, how do I see myself as a Mathematics teacher in East Asia? I am the truly “East Asian product” type as I believe that students must have the Mathematical content knowledge (“product”) before the process developed. That does not mean I advocate root learning. As what Confusion has said “Seeking knowledge without thinking is labour lost; thinking without seeking knowledge is perilous” (?????????????), I strike a balance between focusing on the process and the product. As a ordinary teacher who wants nothing but the best for my extraordinary students (who just have the knack to make me both angry and happy at all times), I used varied teaching strategies based on the different learning theories (behavourism, cognition and construction) to make mathematics both meaningful and fun for the students. For example, I designed varied activies for my students from getting the students to practice their numeracy skills using web resources to using Scratch to explore the geometric properties.