Chapter 9 - Talking and Writing about Mathematical Ideas

     When most of us hear the term math, or if we were to be asked about activities concerning mathematics, not many of us will think of interactive activities.  The text however, suggests otherwise.  Children “need to talk and write about mathematics”.  The importance of basic interpersonal communicative skills being  incorporated into mathematics is that it will help the students make connections to the content.  Similarly, they can get feedback as well as help from one another by communicating to each other about the content.  Ultimately talking and writing about mathematics will help them make meaning to the content.

Chapter 8 - Ungrouped and Pre-Grouped

      When it comes to place value, many students may initially come across difficulty understanding the concept.  Two types of materials the text explains that can help children learn this concept are ungrouped materials and pre-grouped materials.  Some examples of ungrouped materials are beans, straws or more commonly seen are cubes.  These individually are ungrouped.  However, once these items are pout together in clumps such as a pack of ten beans or a bundle of ten straws or a pile of ten cubes, then they are called pre-grouped materials.  I have seen these manipulatives used in classrooms before and I’d like to use them in my classroom as well if I were to teach this concept.  Being able to actually see items and count out the objects in front of you has been found to be very beneficial.  The use of pre-grouped materials in particular gives the students a better grasp on place values for clumps of numbers.

Chapter 17 – What Do You Call Those Lines and Bars… Oh Yeah, a Graph!

            Classic conditioning.  That’s what happened to me at least.  When I see a graph – pie graphs, line graphs, bar graphs – you name it, I fear it.  These visual tools are only meant to help you understand and analyze data.  For example, let’s take a look at bar graphs and histograms.  Bar graphs are used to display separate and distinct data.  The text gives an example where they graph the “number of children’s birthdays in each month or the number of students who travel to school by bus, by car, or on foot.”  If a student were to be given a word problem and had a bar graph displaying this information, the child would have been able to utilize the graph as a quick visual comparison of the data, which would obviously be beneficial to the student in solving the problem.  Reading the explanation for the text of the function of a bar graph really makes sense to me as a helpful tool. However intimidating the concept may be to me, I know one thing’s for certain – I need to strive to show students as the text has shown me, just how helpful these visual aids can be.

Chapter 15 – Geometry

            An interesting point the text made was that children are taught the names of geometric shapes, “but they do not develop the discriminating power they need to use the names with meaning”.  I really enjoyed some of the beginning activities the text suggested.  For example, a very simple and activity that most instructors already use would be “Who am I” activity.  This is where you put out three different objects and describe one of them (it is round all over) and have the children explain their reasoning.  Another activity, the “How are we alike or different” was a nice way to have children learn shapes.  However, instead of simply holding up two solids and have the children tell how they are alike or different, I would incorporate  Basic Interpersonal Communication Skills by having a pair of objects per pair of students have them discuss how they are alike or different to each other.

Chapter 14 - Patterns

            I don’t recall coming across too many pattern questions in the past.  I now know the importance of patterns – it can help children ‘organize their world and understand mathematics’.  There are several kinds of patterns.  One of them is the classic repeating pattern.  My question is, how do you explain a pattern to someone?  Isn’t it self explanatory – you see the repetition in something and there you have it! A pattern!  Let’s say there are objects that follow as such: red blue, red, blue  and it asks what is the color of the following object.  Some children might say red (which is correct) and some might say blue.  Instead of just telling them the answer is red, the text suggests that children have minds that are difficult to get so it is important to have them explain their ideas to you.  Sharing ideas help the kids consider alternative ways to look at patterns.  Studies show that kids should start by recognizing patterns and thinking about them and then eventually encouraged to look at patterns in different ways.

Chapter 6 - Cultural Connections

            The talk of the town has been about how America is “falling behind”.  Of the many ways this statement can be applied, one of the ways is the education or the pace at which our students are going at juxtaposed to those in other countries.  Take Japan for example, the text mentions how Japanese students have some of the highest overall scores whereas American students have some of the lower ones.  This is something I knew beforehand.  What I didn’t know was the difference.  I assumed students generally worked harder in Asian countries therefore higher scores resulted.  I didn’t even think to look at the difference in lessons.  When comparing American lessons to Japanese mathematics lessons, here are the findings:  Americans teachers will demonstrate a sample problem (an exercise, not a ‘true problem’) and then the students will be working on the problems as the teacher goes around to individual students who need help.  In a Japanese mathematics lesson however, they find that the teacher will give the students a complex problem and have them work on their own or in pairs.  Finally, the class as a whole will have a discussion (managed by the teacher) on different approaches to the problem.  What I did not know that is key to this example is that in the Japanese lesson example, the key thing to note is that the Japanese teacher will move around the classroom to give assistance as well as to take note of the students’ approaches to the problem.  The Japanese teacher plans on which students to all on during the discussion.  I would love to try and apply this method in my own classroom.

Chapter 15 (VBS) - The Exceptional Student

            There are four variables that can influence the math ability of students – cognitive, education, personality factors, and neuropsychological patterns all can have a say in how well or poorly for that matter a student performs in mathematics.  Cognitive factors – the intelligence or cognitive ability of the student, educational factors – the quality or amount of instruction, personality factors – the amount of persistence or self concept towards mathematics, and neuropsychological patterns – issues with perception, memory, or even trauma influence are all things that most people agree happen and is pretty self explanatory.  However, an idea that is completely new to myself is one that is concerned with the developmental arithmetic disorder.  This is a disorder I have never heard of but brings about mixed emotions.  First is relief because I knew there had to be some kind of scientific reasoning as to why there are such large numbers of people who struggle with mathematics.  Second came empathy – although I don’t consider myself a person who has the disorder, I can empathize with those who do, because I know the frustrations and the feelings that come with learning arithmetic.  Thankfully we as educators know that there are tools and methods in which we can help those who are struggling with complications such as the developmental arithmetic disorder.

Chapter 10 - What Do You Mean Calculators Aren't Evil?!

            Of the many computational tools out there, I believe the calculator is the most widely known.  From an early age, I recall being looked down upon for relying to heavily upon the tool.  I also recall being in the classroom and being discouraged from using it as if it were a form of cheating – as if I was making the tool do all the work for me.  I do think I rely on calculators to this day – I usually carry one on me at all times.  However to my relief, this might not be such a bad thing.  The NCTM says that “appropriate calculators should be available to students at all times”.  More importantly discussed in the Reys text is that it says calculators don’t discourage thinking and it does not harm students’ mathematics achievement.  Contrary to myths and common belief, the calculator has so many pros as opposed to cons to it such as: it facilitates problem solving, relieves tedious computation, removes the anxiety from computational errors/failures, and it can provide users with motivation and confidence.  Finally, students who use calculators usually have a better attitude towards mathematics.

Chapter 4 - Assessment

            Of the many ways Reys discusses how to gather evidence as to the abilities, dispositions, and interests of the kids in your class such as observation, questioning, interviewing, and many more, self-assessments stood out to me.   This is something I did not know of because I do not recall using this in the past.  The text suggests that students are the best assessors of their own work.  When the students are given the responsibility of justifying their own thinking process, they tend to be more alert because once again, that responsibility is their own.  The text gives an example of this in action: ask your student how he or she came across arriving to the answer to the problem as he/she did.  This method can similarly be used to assess the student as far as how they feel about mathematics – to once again aid the instructor to better understand the student and ultimately help the student better in return.

Chapter 3 - Plan, Plan, & More Planning

What I Already New

What I Learned

·      At the beginning of the school year, you have to decide what your children will accomplish during the year. ·      Most schools prepare scope-and-sequence charts or guides. ·      It’s probably a good idea to double check with other teachers and the principal. ·      When you plan a lesson, try to ask children questions that make them think deeper as opposed to asking them to recall a fact from memory. ·      Really understanding the students and the level that they should be at is essential to planning – getting that wrong will throw off all your planning!

·      Textbook series sometimes come with scope-and-sequence charts or guides. ·      Some schools require teachers to have a lesson plan book where the objectives for each day’s lessons for a week are outlined in detail. ·      Try and incorporate the use of manipulative materials in planning – research indicates lessons with manipulative materials have higher probability in achieving better mathematical achievement. ·      Importance of initiative in learning - learning is most efficient when the student can identify where they specifically need help and then get the help they need.  Plan your lessons to be open to letting students take initiative.

The chart above simply sums up what I already knew or assumed prior to chapter 3, as well as what I didn’t know previously and now know after reading.  All in all – the message about the importance of planning has been received!

Chapter 2 - Creating a Safe Learning Environment

            In the text by Reys, Piaget suggests “knowledge is not passively received; rather, knowledge is actively created or invented by students.”  However, the next question would be how would we get students to actively receive knowledge?  Besides reflecting on their own mental actions and recognizing patterns and what not, learning is suggested by the text as a social process.  Ideally children would be engaging in dialogue and discussion.  To achieve this, I learned or confirmed what I had already known – the importance of a positive learning environment.  This can be done in a number of ways but to mention a few, let’s start off by saying that teachers should be aware of their students and their academic stage – what material is appropriate for which stage their students are in.  This awareness of your own students can help lessen the anxiety of the students and is simply just logically efficient.  Another way to create a safe and positive learning environment is emphasizing communication.  Contrary to my previous assumption that student to teacher communication was the most vital, the text emphasizes not only the student to teacher but also the student to student communication.  Talking between students provides the opportunity for not only explanations, but also for sharing methods, while perhaps simultaneously lowering the anxiety of the student in comparison to a student to teacher scenario.

Chapter 1 - What is Mathematics to You?

              Mathematics.  What a heavy word.  I’ve had mixed emotions when it came to my experiences with mathematics, most of them unfortunately being negative.  For whatever reason it was, whether it was the tedious repetition of text problems or the never ending worksheets or even the frustration from lack of confidence in the material, there are too many components into what shapes my personal definition of what mathematics is to me.  However, one thing is certain.  Of the many things Reys defines mathematics as, such as: a way of thinking, an art, a language, or even a study of patterns, I agree with the idea that mathematics is a tool.  However, mathematics is not just simply any tool.  It is an essential tool.  It’s a tool used at all times in daily life.  Similarly, it’s a tool that needs building.  You cannot go on if the foundation isn’t there.  Like many other subjects, mathematics requires you to master the material before the next in the sense that it builds upon itself.  The point here is, is that as an instructor seeking to have students effectively utilize mathematics as a tool should aim to do what they can to enrich the learning process as it will affect not only the student at their present level but also their future in mathematics as well as their future instructors and so on.