Wednesday, July 1, 2015
All I have to say about chapter 9 is yes, yes, yes, yes, yes!!!
Alright, I have a bit more to say than that but what a great chapter. I found this chapter to be so affirming in terms of why we use word problems and how to teach our student to navigate these tricky, tricky waters.
The chapter discussed students solving word problems in order to learn about and solidify an understanding of operations, to further develop number skills and concepts, and to promote computational fluency. This is to say that a lesson containing word problems could be addressing any number of mathematical goals. This is something the teams at my school talk about a lot. We are using the Engage NY modules but I am sure the conversations we have mirror those of any elementary teacher picking up a program and determining what that means for daily instruction. Questions we often have include "Why are the modules using such BIG numbers! Second grade standards only expect word problems to 100?" or "Why are these word problems so easy? Of course my students can tell how many cookies there are when Ashley brings 3, Sarah brings 2 and Suzy brings 7 cookies... can I skip this day?" Sometimes we are right on the money with our questions and we are able to adjust the lessons as necessary. Sometimes, we are not understanding the purpose of the lesson. Maybe that second grade question isn't really about solving a story problem with sums to 100. Maybe that second grade question is about adding together two three digit numbers but the context of the story allows the students a jumping point for an invented strategy. Maybe the first grade word problem isn't really about adding 3 numbers together, maybe it's about noticing that when combining three numbers, you can be strategic about the order in which you combine them for ease in solving. Ashley's 3 cookies and Suzy's 7 cookies make ten cookies. Add on Sarah's two and we can quickly notice that 10 and 2 makes 12! Whenever introducing a word problem is it important to note the PURPOSE with which you are presenting that learning opportunity.
This chapter also had a small section on avoiding key words. Yes, yes, yes, yes, yes. I wrote a rather long entry on avoiding key words a few months back and this chapter confirmed what I had written... and did it in a much more concise manner :) My favorite quote from this section stated "The key word approach encourages children to ignore the meaning and structure of the problem and look for an easy way out." I could probably go on and on about this quote but isn't it the truth! When we teach our students these shortcuts we are doing them no favors towards helping them to develop their number sense, operation sense, and overall ability to think mathematically. If you are using key words to teach word problems (or rhymes to teach rounding, or "The Butterfly Method" for comparing fractions or any other method that requires no more than a rote application) your students may show success on the lesson today, but will they know it is appropriate to apply this rote process tomorrow? And next year, will they have a strong foundational understanding of operation upon which they can connect to and build future math understandings?
*Stepping off of soap box*
So, if key words are off of the table, how can we help our students to solve word problems? The chapter discussed the methods that students might use to solve an addition or subtraction word problems including direct modeling, counting strategies, and derived facts. Interestingly enough, the authors mentioned that younger children sometimes have an easier time attacking word problems because they think more completely about the context of a problem whereas older students "think that [computation] is what solving story problems means- grab the numbers and compute". A way around this with older students is to require explanations for the way in which a student arrived at their answer. And an equation is NOT an explanation. When students are required to draw a picture or use words to justify their response they can no longer "plug and chug" to solve.
In terms of modeling, the text cautioned students from drawing intricate illustrations in order to solve a word problem. This thought was echoed at a math workshop I was able to attend a few months back. The speaker at the conference really resonated with me in terms of how she recommended dealing with this problem. She said not to tell students that their way of solving the problem is wrong- it's not! She said that if students are drawing a full on illustration that you want to pair their work with a more abstract representation, for example, a student who drew circles to represent bunnies. Ask the student to find the different features in the word problem in their own illustration but then ask them to find the part in the other students abstract representation that means the same thing. Then ask the student which method was more efficient (or faster) when it came to getting the right answer.
The prompting might sound like this:
The story said that first there were 3 bunnies in the field. Where are the three bunnies in your drawing? Oh, yes, that matches the story. Let's look at this model of the story. Can you see the part in this model which shows the three bunnies that were in the field at first? Oh, you see a group of three circles. They used circles to stand for the bunnies! Which way is faster or a more efficient way of showing three bunnies? The next time we solve a problem, why don't you try this more efficient way of drawing.
This conversation does not need to occur 1:1 but could be a full class discussion. Additionally, this method of comparison and conversation can move students from a direct modeling approach to a more efficient bar model or tape diagram as well.
The ends of my posts are beginning to sound like a broken record but, really, truly, this chapter was so dense with information that my reflection is only brushing the surface. I would recommend clicking the links below to read others' reflections on the chapter and/or picking the book up yourself!
at 4:29 PM
Sunday, June 28, 2015
Chapter 8- Early Number Sense
So, let's get into it! The chapter began by discussing early counting and the authors defined early counting as an interconnected set of skills including number sequence, one to one correspondence, cardinality, and subitizing. I felt confirmed when the authors talked about how you can not rush these understandings and students need a high number of activities which allow them to build these skills. Thinking back to chapter 1, it's all about building experiences which allow students to build connections from activity to activity in order to construct a strong meaning of numbers and counting. These activities could include rhythmic counting both forward and backward. Building a number path, counting a number of items, gathering a collection of a given number of items, etc.
Next, we moved into the learning trajectory for counting. As I read the 5 steps laid out, I could easily state which of my students fit into each step and having this understanding will help me, in the fall, to take next steps to move them to a more complex understanding of numbers and counting. The five steps here included emergent counter, perceptual counter, figurative counter, counting-on counter and non-count-by-ones counter. Figurative counter was a step I was not formally acquainted with previous to reading this chapter. Figurative counter refers to a student who, given a set of items with a known part of the set covered, would imagine or visualize the hidden items and would still count beginning at 1 while accounting for these items.
We moved from counting into the number relationships which inform number sense. The relationships which we need to help our students to develop include spatial relationships, one more/two more/one less/two less, anchors to 5 and 10 and part-part-whole. Each of these relationships informs the next and activities in each will help students to develop a robust understanding of the relationship between numbers. I have mentioned, more than a few times before, the book "Fluency through Flexibility" this book is based largely on the research presented in this chapter of the book and relates hands on activities and assessment for developing an understanding of number relationships in our youngest students. I would so highly recommend this book - it's really not very expensive- and you will get so much out of it!
One note in this chapter that I found particularly interesting was the discussion of "calendar time". The authors made the point that using the calendar the promote foundational mathematics because a calendar groups numbers by 7s rather than by 10s. Patterns found in a calendar can be an "in addition to" activity but shouldn't be the basis of a classroom's number relationship discussions.
at 1:21 PM
Friday, June 26, 2015
Chapter 7- Parent & Community Support
Proactive communication to families could come in the form of a family math night, newsletter, videos, a website or other methods of open communication. Get out there and explain the standards you are teaching, how and why you are teaching them. New curriculum or methods can be scary to families and you want to have lines of positive open communication open so that families have a thorough understanding of changes rather than drawing negative conclusions out of fear themselves.
My favorite idea from the text was to present families with versions of a task and compare an contrast the benefits and drawbacks of each. For example, give a list of 4-5 basic addition problems with sums to 10. Next, give parents a problem based task such as "A vase has 7 flowers. Some are red and some are blue. How many red and blue flowers could there be?". Ask parents to discuss and reflect on the skills being developed in each task, which task allows for real world connections and which task is more motivating to the child to solve. I loved the idea of talking about using a problem solving based approach because it supports eventual mastery.
When in a situation where a more reactive approach is needed to a parent or community member who may be unsure about a new curriculum or method of math instruction, the text talks about discussing the fact that the traditional methods of teaching math in our country were ineffective! The text pointed out that our students are average at a 4th grade level as compared to other countries, however, after that point, we fall behind much of the rest of the world. Our students fall behind beginning in 5th grade?! So when a parent says "Why aren't they teaching kids the way they used to any more? I learned the old way and I turned out just fine." A response indicating that, on the whole, our country is NOT turning out "just fine" would be appropriate. Worded in a kind and balanced way, of course :)
If you haven't grabbed a copy of this text, I would HIGHLY recommend it. Each chapter is so dense with information that I am only barely touching the tip of the iceberg in my reflections. I have learned so much that I will be able to implement into my practice next fall! Click here to head to Amazon to check it out.
at 3:16 PM