Forum Replies Created
-
Posts
-
I wonder why we then still have the footnote, “Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.” We can’t ask students to compare 7/4 and 8/5, for example. Or, is it ok if students come up with fractions with denominators other than these in the process of their solutions? In other words, the restrictions apply to problems we give to students…
I don’t disagree with Kipra, but I think we need to articulate what is meant by “strong base 10 understandings.” Then, we need to articulate which part of that strong understanding is appropriate at what grades. Furthermore, I think we should realize that a part of the reason for teaching multi-digit addition/subtraction calculation is to help students deepen their understanding of the base-10 numeration system, as well understanding of the properties of operations – not just having an efficient calculation method. So, by learning how to add two 2-digit numbers, students should come to understand that in order to add two numbers, they must both refer to the same unit. That idea carries through addition (and subtraction) of not only whole numbers but also fractions and decimals. 2nd/3rd grade instruction of multi-digit addition/subtraction should keep that in mind. Another important rule of our base-10 numeration system is that we must use one and only one numeral in each place. That’s the reason we must re-group. So, when we teach addition/subtraction with re-grouping, we are not just teaching an efficient calculation method. We are trying to help them understand how our numeration system works.
From that perspective, I don’t necessarily see anything wrong with start teaching algorithms in Grade 2. Understanding of the base-10 numeration system and understanding of calculation algorithms are intertwined, and they should be taught with their connections in mind.
Bill,
Perhaps my question wasn’t clear.
As you know, a word problem of the type, put together/take apart with addend unknown is like this one:
There are 8 children in the play ground. If 3 of them are boys, how many are girls?
According to the standards, I think this type of problem is to be discussed in K – I am interpreting that is what it means to understand subtraction as take apart. However, the progression document says this type is not to be included in K. As I interpreting the standards correctly?
Bill,
I’m perfectly ok with the idea of children informally discussing about shapes. But, as we look at a K-12 curriculum, shouldn’t there be a time where formal definitions are expected to be understood and students will use the terms accurately – isn’t it a part of mathematical practice? Doug’s quote seems to suggest that point to be middle school or even later, without specifying the timing. As I quickly glance at Geometry standards in middle grades, they don’t seem to specify formal definitions – yet in many ways, formal definitions may be needed for some of the standards. So, I remain puzzled – but that seems to be pretty much a normal state for me 🙂
Bill,
Thanks for relaying Doug’s response. However, his response is very unsatisfying.
How do children know a quadrilateral is a parallelogram? What are the defining characteristics of parallelograms? In theory, we can define parallelograms as quadrilaterals with opposite sides being equal. I can also see a possible investigation by Grade 3 students by focusing on the lengths of sides of quadrilaterals. They can identify rhombi as having 4 equal sides, parallelograms and kites having two pairs of equal sides, etc.. But, this definition, though useful in identifying parallelograms, will be difficult (if not impossible) for 3rd graders to use to draw parallelograms. Trapezoids are even more problematic. When we introduce trapezoids as a type of shape, exactly what are we wanting students to know as “defining attributes” without parallelism?
I think I can rephrase my question like this: at what grade do we say (for example) it is NOT ok for children to say an ellipse is a “circle”? I think it is perfectly fine for K & Grade 1 students to call an ellipse a circle – because they are distinguishing it from the figures that are made of straight segments. But, is it ok for Grade 3 students? Grade 5?
I certainly do not think we should be treating geometry very formally in elementary school. However, we should be introducing formal definitions (stated in language that is appropriate for elementary school students) gradually. Otherwise, elementary school geometry instruction becomes just a series of vocabulary lessons (a bit of over simplification, I realize). I’m afraid when we are so loose about the definitions of figures, we are developing more students who think squares are not rectangles. Children need to understand what makes some shapes a particular type before they can start investigating the relationships of classes of shapes – as they do with 4-sided figures in Grade 3.
