It’s very true that while multiplication often looks and functions like repeated addition but that the operation itself really isn’t quite that simple.  The example of fractions (and also decimals) is a perfect example of that.  When you use multiplication in numbers less than 1 (or even numbers greater than 1 but with a fraction or decimal attached), it becomes clear that it isn’t really related to addition at all.  That brings us to the other main point …

This is probably why math is such an abstract thing to students.  I’m sure this was the case with me.  Just like he describes, I would learn rules or sets of operations only to learn that they don’t apply to something else similar or that they are totally wrong in some cases.  This just made for one big confusing subject for me, and eventually I got to where I had so many different formulas and such in my head that I didn’t know what to use where and obviously didn’t understand any of it.  If we teach students to understand what they’re doing, when to do it, how it works, etc, then I doubt we’d have this problem with the “math-challenged” people like me.  I probably would have been a lot better off if I just had some sort of framework for understanding it.

His last point is that clearly, the change needs to be made with the teachers and the texts first.

While the second article was about sixth graders, it was still very useful and informative, as was the first article (about second graders).  Both articles were a reflection of the teachers on giving their students a problem to solve and examining the methods of the students.  The only real difference between the second and sixth graders was that the second graders were given paper and their choice of manipultives to use while the sixth graders were required to solve it in their heads and then report on their thinking.  The methods of the students were strikingly similar despite the age gap.

There were several different methods used, but nearly all involved renaming, or breaking the two digit numbers into “nicer” numbers that were easier to work with.  The one I identify with more readily was the method of first adding the numbers in the tens place, then adding the numbers in the ones, and adding the two together.  This was the first method mentioned in both articles.  There were many other methods, the second method mentioned involved taking the number needed from the ones place of one number to make the second number contain a zero in the ones place (just another way of making a “nice,” round number).  I think the one that struck me as the most odd was the method of using subtraction to make one number nice and round — that just seems backwards to me; but alas, second and sixth graders thought of it and applied it to make it work for them.  And of course you did have some that claimed they used no renaming at all, such as the sixth grader who said she just invisioned the two numbers vertically and used the standard algorithm, the second grader who just used the 100 board to count by ones, and the remarkable second grader who used coins and drew pictures of the coins needed to solve the problem (their original word problem involved money).

This just kinda reinforces to me JUST how far behind my two kids are.  Granted the are EC and LD in math, but I just don’t see how they’ll ever be able to think like that.  The thing I’m able to take away from these articles and use with my kids is just to remember that students think in different ways and it’s important for me to teach it in different ways in hopes that each of them will connect with one of those ways.  Even if it’s a way that confuses the heck out of me (i.e. using subtraction to solve an addition problem like the kids in the articles), it’s certainly worth the effort for me to either learn to explain it myself to help them understand or find someone who can.  If I can just find SOMETHING that works for them, then I’ve accomplished my goal.

As a side note, I’ve had more success with Touch Math with my kids than I had with standard concrete manipulatives, so if anyone else out there is teaching single digit addition and having troubles, Touch Math is certainly worth looking into.

Last year when I assisted in First Grade, we did centers this way.  At our math center and writing centers, for example, students produced tangible artifacts that were turned in, while the teacher and I each conducted a slightly different reading center where we lead the group in learning and activities and it was easy to assess the individuals by observation (we typically had 3-8 per group).  The students rotated through the centers so that they were at each of our reading centers each day and got to each fun center once by the end of the week.  This kept it fun and interesting for them, but it was not totally without a down side.  The disadvantage to this approach was that the students would often finish the activity at their fun and less supervised center (especially cooking!) and would either come to interrupt a teacher (leading a reading center) and ask what to do with their remaining time, or they would distract the others by either getting loud or directly distracting a friend at another less supervised station like math, writing, or listening.  I feel it would have been more productive to either create a consistent/systematic way for the students to occupy their time if they finished a center early, or do what I am doing this year in my Second Grade class.

In my Second Grade class this year we are doing something called Workshop Way.  Workshop way is somewhat like Centers in that there are a variety of other tasks going on while the students rotate through the small group reading instruction.  It is different in that the students work individually and at their own pace.  There is a task-line posted on the wall (the tasks rotate around and/or change daily) and the students simply work their way down the line at a pace appropriate for them.  There are always more tasks than can be completed, so even the fastest workers never run out of things to do.  The assignments where tangible artifacts are produced are closer to the beginning of the list so that they get done first and the lesser important tasks are toward the end.  This approach has the potential for students to distract each other and be distracted and become loud, but with proper and consistent classroom management, this is easily kept under control.

As I’ve said, I feel that assessment during centers to be an easy task, but while centers are certainly useful, I’m really liking the Workshop Way method better.

I have two big concerns, really.  One is my belief that I am totally inept when it comes to math.  If I’m not confident in what I’m doing, my students will know it.  I’m sure I’ll be fine with the basic math in lower grades, but if/when I ever have to do anything with algebra (and even long division to a point), I’m afraid I’m going to really struggle and as a result, be a crummy teacher to my students.  My other big fear is having a child that’s really struggling and I’m unable to understand what they don’t understand, if that makes sense.  If I can’t figure out what they’re missing, how can I help them?  I know that’s why we have so many steps and ask them to explain their processing aloud – so we can hear their thoughts verbally and try to analyze that, but I still have that fear that I won’t be able to help a child and that I’ll contribute to a negative construct of math for them in their future.

I really need to develop a little more math “ability” of my own, or at least a bit more confidence in my ability to do math.  I also feel like I could benefit from more experience with assessments – exactly how should this be scored and why.  I’m certain that I also need to work on just acquiring a variety of techniques to teach the many different types of learners.  As far as probing for understanding goes, I’m a little concerned that I may not be very effective with that and it’s rather difficult to put “why” into words.  I think I’m afraid of just not being able to pose the right questions in the right way to get them thinking in the direction I need without giving something away … if that makes any sense.  I’m afraid my questioning will end up being either too broad or too specific to gain the greatest amount & level of insight.

Partial credit is a MUST.  With so many steps in some problems, students are bound to miss one of the 4 or 5 steps in a single problem and if they got the entire thing wrong because of that one mishap, I doubt if anyone would ever pass or even WANT to attempt that kind of problem again.  As much as anything else, they really need the confidence boost that comes from seeing that even though they got the final answer wrong, they were on the right track and got it mostly or partially right.  Complex problems definitely require a complex grading scheme.  I can definitely see the point of having a rubric, too.  As much of a pain in the booty as I’m sure it is [will be], it is essential in order to be effective and consistent.

1)       What type of question is being posed?

2)      What level is the child at?

3)      Evidence for #2.

1.       Asked how the child knows which one is what shape and how the child would explain to someone else how to find that shape.  The child is at level 1 because her only explanation for how to identify a shape is to say “It looks like this …” and draw it in the air.

2.       Asked child what shape it was and how he knew that.  I would still say this child is at level one OR maybe between levels 1 and 2 because although he was able to describe the triangle to some degree (“has a point at the top and is flat on the bottom”), he became confused when the instructor oriented the triangle to be upside-down.  At first he said it was no longer a triangle because the flat part was at the top and the point on the bottom.  When it was turned sideways he said “yes [it is still a triangle] … but it’s a sideways triangle.  When it was turned back upside-down, he the identified it as “an upside-down triangle.”  Although his ability to describe parts of the triangle would place him in level 2, I think his easy confusion based on orientation holds him still in level one or possibly between the two.

3.       Same child as in the second video and was given a differently shaped/proportioned triangle and asked what shape it was.  I think this clarifies that the child is indeed still in level 1 because he was unable to identify the shape at all.  Because it did not exactly match the isosceles triangle as before, he could not identify it as even being similar in terms of the number of sides and angles.

4.       Asked about classifying similar and congruent shapes.  This student is at level 3 because she can give a clear and accurate definition of the terms, she was however a little stumped on being able to identify that shapes that are similar are not necessarily congruent, but if shapes are congruent, they must also be similar.  The abstraction is there, but the deduction isn’t quite there.

5.       Asked about angles made with two wheels/circles.  I think this student is at level 4 (although possibly 3 just because there may not have been enough information to go on) because for the most part she answered clearly and without hesitation that the two differently sized circles set at right angles had the same size of angle because the degrees were the same and the size of the overall circle was not important.  She also identified that there are many different sizes of acute angles and that again the size of the angle was independent from the size of the circle.

6.       Child arranged the wooden shapes to make letters from some and houses from others.  I’d say he’s at level 3 because he has the abstraction down and identified that the cube was a “3D square” and that the two are different because “squares aren’t 3D.”  He argued his points/ideas well enough but there was no evidence that he could be at level 4, so I would keep him at a level 3 ranking.

7.       This child was apparently asked to arrange the triangles in a way that was meaningful to him and used them to spell his name and then made a star and a dog out of the remaining triangles.  I honestly don’t feel there’s enough information to assign a level to this student.  The administrator basically just acknowledged his arrangement and did not ask any other questions that would be of value in establishing what level the student is functioning at.  I could be wrong, but I just don’t feel there is enough information to form an opinion.

My performance on this task (individually and in respect to the rest of my group) was typical of similar activities in Dr. Piel’s class.  I picked it up pretty quickly in terms of how to do it and then just had to articulate it accurately to my group.  This still always strikes me as strange because I’ve gotten so many of these tasks so quickly and easily and then helped my group to understand them when all through school I was slow in math and struggled a great deal as compared to my other subjects.  Now if we have to deal with Algebra, you’ll see my typical performance in math; but I digress …  I was able to see the pattern of the blocks fairly quickly and then helped my group to make sense of it. 

We discussed our solutions and approaches with the class and also analyzed student work and explanations of similar things.  We looked at their answers and explanations of work to see who had the better understanding of what they were doing and created questions to ask them to aid and/or enhance their understanding.  That’s pretty important because it helps the slower students get to the right place but it can also be used to enhance the understanding of the quicker students both so they don’t get bored and so they may be able to help their classmates.