Guided Practice
Today's Activities:
Today's Activities:
1. Solve a math problem and reflect on your metacognition process.
Guide:
a. Read the following math problem (we are not looking for the answer, it is okay if your answer is wrong. What we will need is a reflection on your problem solving process and strategies).
b. Before you solve it, think: What do you know about solving this problem? Have you solved a problem like this before/recently? What strategies could you use?
c. Solve the problem. Think about your thoughts when solving the problem.
d. Reflect. If you solved it quickly, what worked? If it took you a little longer, what strategies did you use? Did you stop to change your strategy? Were you conscious of your thinking while solving the problem? how? what were your thoughts? If you did not get to the answer: what do you think did not work? What knowledge or experience are you missing that you would need to solve that problem?
Problem:
Kara wanted to earn a 85% for her overall history grade. During the course, she has to take 5 tests. So far, she has only taken 4 tests. Her exam scores for the tests are as follows; 85, 83, 70, 95. If Kara wants to ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
Kara wanted to earn a 85% for her overall history grade. During the course, she has to take 5 tests. So far, she has only taken 4 tests. Her exam scores for the tests are as follows; 85, 83, 70, 95. If Kara wants to ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
For Today's discussion:
1. Using the guide above, describe your metacognitive process during your problem solving.
2. After evaluating your own metacognitive process, how do you think teachers could help students think about thinking when solving problems? You can refer to p. 255 for tips.
Well I broke down the question. (With my southern accent ;0)
ReplyDeleteFor a perfect score it would be 500pts, but since she is only looking for an 85...
I would divide 500 by .85 and that would give me the total points needed to get 85%.
It turns out that she needs 425 points. If I add up all the points now... she has 333pts. Subtracts what she needs 425points from what she has 333 points and it will tell her what her next grade must be 92. She can check it by adding all her old grades and the 92 and dividing that answer by 5...she gets a perfect 85%.
Well when thinking about math word problems, you have to break them down step by step and don't move on until evevryone understands how you got there. Yes you do need to reach into your schema and possibly draw it out, but it is well worth it. Then I would model again and then have them work in partners as they explain to each other how they figured it out (think it through) then I would have them solve one by themselves showing me the process and explaining it to me. Takes time, but they get the hang of solving multi step word problems.
-Angelica
Kara wanted to earn a 85% for her overall history grade. During the course, she has to take 5 tests. So far, she has only taken 4 tests. Her exam scores for the tests are as follows; 85, 83, 70, 95. If Kara wants to ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
ReplyDeleteMinimum she has to get on her 5th test is a 92%.
First I needed to know prior strategies....in this case I multiplied 85 times 5=425
Then I needed to subtract the number of points already earned (333) from 425 which evidently gave me 92 needed points to get an average of an 85%.
Finally, I used the answer I computed and added all five test 85, 83, 70, 95, and now missing grade 92 and divided the sum by 5 which gave me 85% as average.
I tend to solve and then substitute my answer to make sure the answer was correct.
I tried to comprehend the question by fleshing out the variables and analyzing how each one connects;
ReplyDelete• who-what-when-where-why-how – recognizing that some variables are rarely significant to arriving at the solution;
• looking at the test as the main variable - how many; in what subject – although not as relevant;
• looking at the numbers (what are they and how are they connected and why); and
• decoding the main question stem - interpreting/rephrasing it in my own words so I can “own” the question and I can dictate and phase my own understanding: e.g. “ensuring Kara receives an 85%, what score should she get on the 5th test?”)
*I am NOT a Math person (Math and I are like oil and water! ☺), but it helps me solve word problems better when I break them down into steps.
As classroom application, teachers can use metacognition by modeling the thinking process themselves, and by ensuring that the instruction they give their students are broken down into parts, instead of just one big set of steps. In the actual performance of the task, it helps a lot to give students some time to finish one step before giving them the next as it gives them the chance to complete one and correct their mistakes, if any, before proceeding to the next. As Gagne’s theory denotes, there is a hierarchy to performing a task; and in this hierarchy, prerequisite knowledge (the step before the next) is important.
Vir,
DeleteI'm right there with you. Math and I have never been friends :) . I also have to break down the problems to be able to understand them and make sense of them and that's the way I taught my students when I taught math.
It has been my experience, when we break math problems into steps and reflect their thinking process this really helps English Language Learners understand what it is the math process that they have to follow.
When I do math word problems:
ReplyDeletea) I read the problem slowly as I search for what it asks me to solve. In this case I need to find out what she needs to get on her 5th test to ensure she gets an 85% average for her class.
b) After I know what it is that I have to solve, I start recalling math strategies (if I know any). Since I do not know any math strategies for this problem (I’m not a math teacher) and I do my math the longest way ever, I start playing with numbers.
c) I add all the scores and add a 90 (85, 83, 70, 95, 90) and get an average of 84.6. Second attempt: I add 85, 83, 70, 95, 92 and get an average of 85!. So she needs to get a 92 on her 5th test!
d) Since I don’t know any fast math strategies, I did the math problem the long way. However, I need to look for strategies so that I don’t take too long inserting different numbers.
Alicia, your strategy resembles mine. I teach math, but not to this level. I am sure that if we did we would have a more structured strategy and could solve it faster. But this works for us, we can get to the answer. One important thing is that we can reflect on what skills we are missing and we could work on them if we needed to.
DeleteIf it’s a problem I have solved before or that I know the steps to I always try the shortest way. In this case I don’t remember the steps in solving this problem so I would do what seems common sense to me and start the longest way. Bust out the scratch sheet of papers and my few math skills ;) ..
ReplyDeleteI would begin by looking at what information is given to me in the problem, what information is needed and what information is useless.
I would then get all the numbers that I will be manipulating written down. In this case, the test scores so far.
I would plug in an 85 to the test scores and divide it by 5 to get an average. If it’s not close to the average that is wanted I would plug in a different number and so forth until I get the average needed.
Once I learn the steps and become familiar with them I would take the shortest process possible.
Teachers need to remind students to break down the question into what it is asking for. I don’t teach math but I’ve tested many students in the upper grades and helped my niece with her 4th grade math homework and they plug in useless information that is in the question; resulting in the wrong solution.
1. After reading the problem and understanding the big idea, I make sure I know what the question is, exactly what is the problem asking me to find. In this case, we need to find a possible score for the fifth test to ensure the desired average, 85%.
ReplyDelete2. Now I need to analyze and organize the information the problem is giving me, what do I know, what is the information I have, what is the information that I need. I write it down, I am definitely a visual learner, I need to see it! That will also help me to manipulate the info in any way so I can get the answer correctly (I can lose info if I only use my brain for calculations ;) ).
3. When I have the needed information to solve the problem, then I use my background knowledge and recall information about averages. This is very important, I need to get from my long-term memory the information about what average is and how it is calculated. If nothing comes up, then we seek help from a more knowledgeable source (peer, teacher, book, internet).
4. Now that I recalled how to obtain an average, I use the available information to identify the missing part of the problem. I know that all the test scores need to be added and then divided by the total number of tests to obtain the average, I write the numbers to represent that process.
(Test1 + Test2 + Test3 + Test4 + Test5) / 5 = Average
Substituting:
( 85 + 83 + 70 + 95 + ? ) / 5 = 85
5. It’s time to calculate or solve the mathematical operation (I like this part!)
( 333 + ? ) / 5 = 85
( ( 333 + ? ) / 5 ) 5 = 85 * 5
( 333 + ? ) = 425
( 333 + ? ) – 333 = 425 – 333
? = 92
6. Answer found! Finally I ask myself how do I know the answer is the correct one, how do I make sure. To make sure it is correct, I substitute the answer in the original equation!
Problem solving is a task in which many cognitive and metacognitive processes are needed. Students must be taught all these processes, through modeling and many think alouds. It’s funny, I always remember this Math teacher I had in 10th grade. She used to fill the blackboard with the problem and her calculations, she talked while writing and we were so busy copying everything before she started erasing everything to continue writing the solution (it was a chalk board). Oh, she was sooo knowledgeable, but she lacked the knowledge of instructional strategies to ensure the students understanding. I learned so many things from her, not necessarily Math concepts though!
So funny Silvia, I had teachers like that too. If she had made you listen to her thinking process (and mediate your classroom to reflect on it) instead of just having you copy while she was solving it, you would probably think differently about her. Some teachers are very knowledgeable of their subjects but are not very good at delivering it!
DeleteThe question was asking me to find an unknown number. I knew I need to first add the scores: Find the sum of 85+83+70+95 = 333. I proceeded with determining the total score needed if the average was to be 85. So I multiplied 85 times 5. 85 x 5 = 425. Because I know that to get an average will will need to divide the total. Knowing that the total scores need to be at least 425, I then found the difference between the actual score 333 and the needed score of 425, 92. Getting a minimum score of 92 will get Kara the 85 she wants in class.
ReplyDeleteAs I looked at what I wrote, I saw that I skipped some small steps and some mayor ideas. I don't mention that I would do a lot of this grade game in college to determine the final grade I needed to maintain my grade point average. I had prior experience in question similar to this one. I also did not mention that I use the inverse operation of division to find the total points needed in order to get the desired average. If I look at my thinking, it involved too many step to try to list. Maybe now I can see why my students get frustrated when I expected them to show their work or explain their answer. But I also helps to reflect on what you know, or how the process could have been made simpler.
Problem:
ReplyDeleteKara wanted to earn a 85% for her overall history grade. During the course, she has to take 5 tests. So far, she has only taken 4 tests. Her exam scores for the tests are as follows; 85, 83, 70, 95. If Kara wants to ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
OMG! This is not my Flow Time!!!!!! I do not like math!!!!
Okay I can truly be very honest the only way I have ever gotten through math is by placing a picture in my brain of what the problem is asking for.
1. I am reading the problem I am thinking, thinking, thinking and saying how am I going to get this done! So I am drawing! I draw the little girl and five tests with a grade on four. On the last test I put a question mark.
2. Now I am thinking about the big question mark what grade has to be there to get the 85%
3. Now I am trying to solve .... I multiply 85% x5 because its 5 tests to equal to 85%
Then i have total of 425 so I subtract it from 333 which gives me the 92 that Kara needs,in order to get the 85%.
,
4. Now I reflect by checking my-self I add all five tests with the 92 included and it gives me a 425 then I divide it by 5 and it is the 85%
1. Using the guide above, describe your metacognitive process during your problem solving.
ReplyDeleteI thought about percentages, adding, dividing, and guessing a grade. I automatically added her grades, then guessed a grade more or less on the average and with a 90, she would get an 85 when it was rounded off.
85+83+70+95=333+ 90=423/5=84.6(rounded) = 85
2. After evaluating your own metacognitive process, how do you think teachers could help students think about thinking when solving problems? You can refer to p. 255 for tips.
Teachers can help students by identifying whether the students need pre-requisite skills to solve problems. If so, then guide them to identify those skills. Using prior knowledge and relevant situations will help students, and have students collaborate with each other to find and communicate about multiple ways of solving.
While exploring further, I encountered this web-page that has metacognitive strategies and what to take into consideration when setting up your classroom etc.
http://www.benchmarkeducation.com/best-practices-library/metacognitive-strategies.html
Thank you for sharing your thinking process. Collaboration on multiple ways to solve a problem is a great metacognitive strategy, specially if we help and guide them to express their thinking process. Thank you so much for sharing the page with us. It has really good sentence stems that can serve for metacognition conversations in the classroom.
DeleteThe first thing I did was read the problem slowly so I could understand what the question is asking. Then, I got the important information, which are the tests scores and the 85% overall. After that, I added the tests grades: 85+83+70+95 and it gave me a total of 333.
ReplyDeleteThe next thing I did was add a number greater than 85 and divide it by 5 to see if the average was 85.
85+83+70+95= 333
333 + ? = ? /5 = 85
333 + 92 = 425 / 5 = 85
Teachers need to remind students to pay attention to the important information that is provided and use colors, draw circles or box that information so they don’t forget about it. With my students I always give them colors so they can underline the important information and circle the key words. I noticed that by the end of the school year they were doing it on their own. Going back to the topic of solving a problem, we also have to remind them of breaking down the problem and doing all the steps so they can find the right answer.
Conduct a cursory review of the problem (scan the problem for a general understanding). Outline the facts (determine the necessary information, disregard irrelevant information). Make a plan (choose the appropriate numbers, operations, sketches, tables, etc.). Check for reasonableness (do the operations make sense, estimate the solution). Solve the problem (do the math). Check for accuracy (review, does the estimate reasonably match the solution? Use strategies to check the math).
ReplyDelete
ReplyDelete85+83+70+95+n= ?/5=85 [according to the problem the minimum score (n) would have to be added to the four other scores, the sum would be divided (by 5, the mean) and the quotient rounded up to show an average of 85% in the course]. 85+83+70+95+n=423/5=85 So, n=90
test
ReplyDeleteSince I am a math teacher I think I needed something different than that. At any rate, I first focused on what I was asked. (Find the value that will give you an 85% average) I proceeded to create a table and did the math to figure out what the percent on the last assessment should be to still earn an 85% average. This was not too difficult for me and it required little thinking.
ReplyDeleteWork:
First I created a table:
Test Score
1 85
2 83
3 70
4 95
5 x
I then created an equation: 333 + x
---------- = 85
5
I then proceeded to solve the two step equation which gave me x = 92%
To solve the two step equation I first got rid of the 5 so I multiplied both sides of the equation by 5 then I subtracted the 333 from the product of 85 and 5 which was 425 which gave be 92.
Math is a subject that requires you to have a foundation (basic skills) because what one learns in kinder is the foundation for 1st grade and as we go through our educational journey we keep building on that foundation. The only way I can suggest for teachers to help students is to create a series of steps that will serve as a guide to help with solving math problems.
I use the following Math Problem Solving Strategy
1. underline the question being asked
2. circle important information
3. cross out trash
4. write down math tool
5. solve using math tool
6. finish your story with a sentence
Now for number 6 I tell them story because I remind my students that Math always tells a story and so finish the telling the story with a complete sentence.
This really helps my kids. Given them something to follow will always give them a head start of what needs to be done.
Esmeralda
Thank you Esmeralda, I can tell you are a great Math teacher. As I was reading how you solved the problem I was thinking, o yeah! that makes much more sense than the way I did it. This is another way we could help our children use their metacognitive skills during problem solving. Allow them to expose the different ways they get to a problem. Not only by sharing their strategy, but also sharing their actual thoughts. This way children are exposed to different thinking systems. Thank you so much for sharing!
DeleteWhen I first approached the problem, I knew that it involved an algebraic equation and mean. I have solved problems like this before. I thought that I would set up an algebraic equation, use the variable “x” and solve for x. While I went through the problem, I made an error and I told myself to slow down because I was doing it too quickly. The reason I knew I was wrong was because I arrived at an answer that did not intuitively make sense. The algebra strategy worked. I didn’t have to change my strategy; I just had to implement it more carefully.
ReplyDeleteThink: What do you know about solving this problem? I know I have an unknown to solve for. I know how to find an average of a series of numbers. Have you solved a problem like this before/recently? Yes, not recently. What strategies could you use? I could use guess and check. I could set up an algebraic equation.
ReplyDeletec. Solve the problem. Think about your thoughts when solving the problem.
First I need to know her current average, so I add her test grades and divide by 4.
85+83+70+95 = 333/4 = 83
I know that one way to get an average of 85 with 5 numbers is to add 85 5 times and divide by 5: (85*5) = 425/5=85
Finally, to find the minimum grade she needs for her series of 5 test scores, I subtract 425 (the "pure average") - 333 ("her average) = 92
d. Reflect. If you solved it quickly, what worked? If it took you a little longer, what strategies did you use? The first one that came to mind was guess and check, but I knew that was not the most efficient one. Did you stop to change your strategy? Yes. Were you conscious of your thinking while solving the problem? Yes. how? I repeated my thoughts a few times, told myself to back up and start over, told myself to think hard cause it's late and I'm tired!
Thank you for trying even though you were tired Carrie, I understand, I am very tired too! :)
DeleteAfter reading all your responses, it is evident that everyone has their own strategies for problem solving. Your thinking process differs by how much experience you have on solving this kind of problem. Esmeralda Andrade for example, shared she is a Math teacher and how this was a very easy task for her. She created a table, an equation and even shared with us some strategies. For others (like myself), it is a slower process, we think, try, re-try, divide, subtract, estimate, go back to divide etc. My thinking process for solving a problem is not a structured as Esmeralda's but based on my skills, it is what works for me. However, one important pattern, was that we ALL used our metacognitive skills. We defined the task based on the strength or weaknesses of our skills, we came up with a plan, we used our planned strategies and reflected on what was working or not (even if it was that we were to tired to focus on the problem at that time!). If we would have to solve problems often we would accommodate our strategies for future problems.
ReplyDeleteMetacognition is crucial for problem solving. So, how do we make sure our students use metacognitive skills when solving problems? Some of you provided very good strategies such as braking the problem in parts, having a structured strategy (1. underline, 2. circle, 3. cross, etc), etc. In addition, a conversation (which can begin with the activation of prior knowledge) can be mediated in the lesson throughout the problem lesson. You can pair students to solve the problem together and tell each other their possible strategy out loud. You can also have students or groups come out to the front to share how they solved the problem and what their thinking process was, kind of what we did today :)
Maria Castillo shared with us a link with metacognitive strategies, it has a good list of sentence stems we can use for metacognitive mediation in our classrooms:
http://www.benchmarkeducation.com/best-practices-library/metacognitive-strategies.html
Have a good night!
I first looked at the problem and separated the variables. I didn't want to be distracted by information that is not necessary to solve the problem. I would analyze the information. So I was left with:
ReplyDeletewants 85% overall grade.
Must take 5 tests.
So far taken 4 tests.
exam scores follows; 85, 83, 70, 95.
ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
At this point I have to retrieve my prior knowledge regarding averages and percentages.
I start to think if Karen wanted a perfect score she would have to accumulate 500 points. I know that if I multiply 500 times 85% I would get the amounts of points needed in order to get at least an 85. I then add her 4 tests to get 333. I subtract 425-333 which equals 92.
Coming up with this answer took me a while to come up with. I am not a math teacher. For the past 3 years I have only been teaching reading, language arts, and history so I haven't had to think about math. I wanted to guess and check but I knew there was an easier way but just didn't know what it was. Once I retrieved my prior knowledge, I was able to figure it out.
An important aspect to help students is to look at math word problems in the same manner we do reading problems. I know when you look at the problem it looks long but once you take out the "trash" then its a much easier problem.
a. First I read the problem twice. As I was reading I was trying to disintegrate the problem by carefully reading the different parts of the problem. In my head I was already thinking of the way I was going to solve it.
ReplyDeleteb. Since I had solved similar problems, I knew what type of strategies I was going to use. I added the 4 scores and divide them by 4 to get an estimate or percentage of the 4 and since I knew she wanted to get at least 85% as an average, I started playing with numbers, I knew the score on her last test had to be greater than 85 because with the 4 scores she had a score of 83% which is lower than 85.
c. It took me sometime because I had to go the long way. But I did not change my strategy because I knew that by adding dividing I was going the answer.
Problem:
ReplyDeleteKara wanted to earn a 85% for her overall history grade. During the course, she has to take 5 tests. So far, she has only taken 4 tests. Her exam scores for the tests are as follows; 85, 83, 70, 95. If Kara wants to ensure she will receive an 85%, what is the minimum score on the 5th test that she can receive?
I taught 4th grade Math this past year, but this wasn't our level. However I'll use what I know about this. I believe this is finding out an average.
Right now she has 333 points (I added the 4 test grades) and subtracted that from 425 points (85 X 5). The difference is 92. I checked my work by adding the 4 test grades, plus the 92 and divided by 5. My quotient is 85, which is the average that she wants.
I had to think back to what I know about calculating grades and I used a scratch paper to work out my problem.
I worked with students on STAAR Math preparation and working out word problems was one of the main struggles that my students had. Many did not have the background knowledge and many more struggled because of the actual reading of problems.
Marie I. Hernandez