We had a discussion before in the math class about memorize the multiplication table. We all know that if the children just know how to memorize the table, they will not be able to explain why 5 times 5 equal to 25. They will not truly understand the meaning of multiplication. But by the end do we need to help the students to memorize the multiplication table? If they do not memorize it, they will have to rely on the calculators or the multiplication table all the time. They will not be able to calculate things fast. How can they survive in this competitive world? Growing up under Chinese Math education system, I feel that I learned a lot of math skills that I will not need during my high school years; teaching in the U.S school, do we teach our students enough math skills?
posted by Guangqiong Zheng at 5:14 AM
12 Comments:
I believe that it is important for children to understand the concept of multiplication. If they do not understand it, how will they be able to apply this knowledge outside of the mathematics classroom? Most of us probably learned our multiplication tables by memorizing. I remember we were given a test every week on the multiplication tables 1-12 for each number through 12. I definitely used memorization to learn multiplication facts, however somewhere along the way I also must have learned how to apply this knowledge and what exactly it means to multiply 3 by 4.
I think it is best to incorporate both styles into learning multiplication. Students should first be introduced to the topic and allowed to experiment with it. Once they are able to understand this concept, they should be required to memorize the table facts. Students will be tested on these facts throughout their lives and should be able to recall them quickly. High school entrance exams, SAT's and eve GRE's will require students to recall and apply multiplication skills. I would have to say that in the US, teachers need to engage in both teaching the skill and apllying the skill by providing students experiences to manipulate and memorize the facts.
I think this question is basically a question about the relationship between understanding and memorization.
Knowledge is not equal to understanding and understanding rather than knowledge should be the ultimate purpose of curriculum. According to Grant Wiggins and Jay McTighe’s book Understanding by Design, understanding includes 6 facets: explanation, interpretation, application, perspective, empathy and self-knowledge. A multicultural curriculum should consider students’ prior knowledge and cultural background and give them opportunities to relate what they have learned in the classroom to their own lives.
Understanding is based on knowledge and memorization is necessary for knowledge accumulation. For example, we have to understand what is good writing and how to do good writing, but if we do not memorize vocabulary and syntax, we cannot write at all.
A lot of math skills do not have many obvious uses in our life if we do not work in cartain fields. However, I think learning math has a gradual and unconscious effect on students that is leting them think in a logical way.
I really struggled with memorizing the multiplications tables. I was a terrible math student in 3rd and 4th grade because of it. By fifth grade however, I finally began to remember them and my math skills improved because of it. I practiced my multiplication facts prior to that all the time. I had flashcards, I did timed exams, I played math games, but nothing seemed to help the facts stay in my head. I finally learned them after three years of intense memorization efforts. Because I did not seem to have memory problems in other fields (except spelling), I believe that my problems stemed from not being ready developmentally to memorize the multiplication tables. I think that the current research's opinion of developmental stages is that students do indeed go through them, but at different paces and ages, and they can be affected by teaching in the zone of proximal development. That being said, while I agree with Cristina that the multiplication tables are important to memorize in order to solve more advanced problems efficiently and get through timed standardized tests, I think we should be careful of student's stages of development and not be too stringint about the memorization too early, unless we are willing to put considerable time and effort into scaffolding the student's developmental levels.
Having been educated in both the U.S. and Taiwan as a teenager, I think the math curriculum in the U.S. is a decent one. Although every state has a different math curriculum, I think overall it teaches students fundamental knowledge of mathematics. I like how students learn sequentially so the difficulty and the complexity of the questions increase as students move to higher level classes. Students who are more advanced in math can have the options to take college level math courses in high school, join math club to practice challenging math problems or compete with other students.
I think the problem lays to the students’ attitude toward learning math. Learning math is like learning anything in life that takes tremendous practice for anyone to get better at it. I feel many students don’t remember the concepts, formulas and the type of questions that they’ve learned before; therefore they don’t remember how to do the questions on the tests. Also I feel students depend on the use of calculators too early. There are many math problems students could do them faster mentally but many students will need to take out the calculator and punch in the numbers to get the answers. I think students should acquire more solid mathematics skills before allowing them to use calculators in math class. Calculators should be used as a supportive tool; students should not feel like they can’t do the calculation without them.
I think that we do not need to have a lesson on how to memorize the tables, but we do need to give the students many activities to practice the tables, therefore reinforcing internal pattern of the tables. I think there's a difference between memorizing the tables and internally knowing the tables because when one memorizes the table if one does not practice they may forget; where as internally the student will just know the answer without questioning whether is correct or not and that is when they always rely on the calculator or the person next to them. I think that in U.S. schools only teach good math skills when students are in a high level/AP math class or in a private school. (From experience).
Memorization has been one of the reasons I have always felt uneasy when it comes to math. I have always relied on what I have memorized, but I always find myself double-checking myself. I feel the fact that I know how to re-work out a problem is very valuable because it is in that manner that I can assure myself of a right answer. I agree to a certain extent that it is important to memorize important concepts of math (like the times tables, formulas, etc.) but we must also focus on the PROCESSES. As educators we much focus on HOW we teach our students. For example, we might want to focus our attention to drawing out a problem and working it through (modeling). This will not only help students visually as well as cognitively. Students must understand the significance of mathematical symbols in order to understand how they will operate in a problem. If we concentrate on the PROCESS our expectations will be met. Once this occurs, we can proceed to introduce memorization, but only as a technique. I agree with Ana-Stephany that we do not need to have a lesson on how to memorize the times tables. What we do need to provide as educators is students activities in which they can implement the process, importance, and significance of practicing tables and mathematical formulas. Through this method we will not be reinforcing internalized memorization, but building blocks that will help them in the future. There is a difference between memorizing mathematical concepts and actually comprehending them. Through memorization we might "know" it superficially, but if we have a deeper understanding of mathematical concepts we will know it internally. Otherwise, as Ana-Stephany mentioned, if we do not practice what we "memorize" we might easily forget it.
My take on memorization is that it is most effective after understanding the concept, in which the product that you are memorizing is a part of. For instance, you may have a student memorize multiplication problems using flashcards, but there will be a limit and the time that it will take them to memorize all multiplication problems may be a lifetime. Unlike, if we were to teach them why 2x2 equals 4, will help them know how to solve for 222x222. I think that this is the most prevalent problem in mathematics in general because there tends to be this urgency for simply getting to the product with no attention to the process and how it may apply to more then one scenario. As for, if we are teaching enough math, I am not sure, but what I am sure of, is that understanding the overarching applications of concepts is much more effective then simple memorization. One may argue that memorizing is much easier, but on the contrary I think that it is harder and takes longer to memorize an entire multiplication table then it does to fully understand one multiplication problem and solve for it.
Teachers need to help students understand the concept, just as parents and students need to review the information at home.
I think that many teachers forget a major concept when it comes to learning: once a subject is learned, it is internalized. The mind is able to retrieve this information easily once it is internalized, and, therefore, recall is fast. This means that most people do not have to rely on calculators or multiplication tables all the time, but the way in which addition/subtraction/ multiplication/division are taught does affect how the information is retrieved when the person wants to use it in the real world. Also, many people can function easily in the world without taking or understanding high school math. However, the fact remains that students take classes prior to high school that discuss those concepts (patterns and algebra for example). People find ways to survive, even in this competitive world. Right now, I think that it is hard to teach for the skill because of testing requirements. Teaching for the test is easier than teaching for understanding. Yes, many math skills are taught in US schools, but many times the way in which many students are taught is lacking because of the aforementioned restrictions.
Students are often forced to memorize things without understanding the concept; I don’t think this is necessarily bad. If a student fully understands a concept, it is likely for him/her to remember that concept without putting in much effort. Thus, teachers should aim to promote as much understanding as possible. However, some individuals experience difficulties grasping certain concepts no matter how hard teachers (or the students) try and some concepts must be learned. In this case, it might be good to have the student memorize what they need to memorize first and provide as many situations as possible where what they have memorized can be applied. It is not uncommon for students to just “get it” after going through repeated process of applying what is memorized.
I think this is a very interesting thought to ponder. As I teach my students new algorithms and concepts I realize that they are given a lot more of an explanation as to why than when I was in school. To this day there are things I learned that I’m still unsure of when I’ll use it in life, such as why we learned about proofs in ninth and tenth grade and how to apply that to the real world. Why are they solved that way? How do we come up with that algorithm? Understanding why something is solved in a particular way or why four is the product of two times two is definitely helpful in many ways. We can actually understand it, not just know it… we can use it to get more than just an A on the next text, we can build further learning off of it. However, if I had to stop and think about what I was doing every time I needed to multiply two numbers rather than just being able to associate an answer with every math fact, it would take me way too long to do any further math and I would have to give up and use a calculator. I think we do learn how multiplication works before we actually memorize the tables, such as when we are grouping objects or skip counting. Right now I am teaching students about equivalent fractions. I know that they could easily learn this within twenty minutes if I just declared that for all the problems on that sheet all they had to do was multiply both the numerator and the denominator by the same number, but instead I had to spend three days showing them with pattern blocks and other manipulatives why these fractions were equivalent. Will this help them in the long run? It’s hard to say if it’s necessary at this point, but I think it does make math a lot more useful to them.
Looking back on my own personal math education, I feel that I did not receive practical information in terms of math lessons. Math was heavily taught yet isolated with no context to other subjects. For this reason, I never found math interesting or relative. Could I memorize? Of course. Did I know how to apply what I way memorizing? No. Math was a class that required sitting an isolated desk doing individual work. I did have one teacher, Ms. Hanley, in the fourth grade who did Math Jeopardy. I remember it was only once a month but loved how it was competitive and interactive- though I never wanted to be called on. Although I do not currently teach isolated math, I believe I incorporate it through cooking in the classroom. I have also observed many teachers at my school, teach math very thematically. Also are more actively engaged in applied math-children are constantly building things while learning about shapes, jumping and measuring to learn about distance. I always though math was about numbers on a page and never saw it in relation to my own body in my elementary years. I think skills are being taught and skills that apply are even more beneficial.
I think the point is that we should not JUST require students to memorize the times tables, because they will have no understanding of what they are doing. In the end, they WILL have to memorize the times tables because, just as Guangqiong says, they need to be able to calculate quickly.
The reason children should understand multiplication before being asked to memorize the times tables is that they can be provided with motivation to learn them. If they don't know what they are doing, it would be difficult to feel motivation.
My story about learning the times tables: I cheated on multiplication tests with an eraser that had the times tables for 1-12 printed on it. At one point I realized what multiplication was - a short way to add groups of things. If I didn't know the answer, I could at least draw these groups and count. After enough exposure to them, I began recalling them more quickly (much as we internalize sums), so I automatically began memorizing them, in a sense.
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