Monday, December 8, 2014

iHeart Math Holiday Hop

Welcome to day #9 of the iHeart Math Holiday Hop! I am so excited to join 22 other math bloggers to bring you holiday tips and treats through the month of December! If you have never been to my blog before, I am a math interventionist in NYS. I work with students in grades K-5 and have a background in intermediate special education. Image Map
Holiday Tip #1: Giving Back
It is easy to get caught up in the stress of the holiday season. I love the idea of committing to 1 act of kindness towards someone else and 1 act of kindness towards yourself each day of the month to keep yourself and others sane! This may be as simple as committing to leaving within 45 minutes of the end of the school day to get home and spend time with family, treating yourself to a special drink in the morning (Starbucks!), or putting down your pile of grading so that you can go to the gym. Work will always be there, it will NEVER truly be "done" so take time for yourself. An act of kindness towards others could be as easy as leaving a copy of that great lesson you are doing on a colleague's desk, bringing in a muffin for the teacher who comes flying in having already readied their 3 kids and dropped them off at their own respective schools. You know how exciting it is to have an anonymous treat dropped off in the morning- commit to giving someone else a great start! 
Holiday Tip #2: Math Tip
Picture this, it's the day before the holiday break, the kids are excited and bouncing off of the walls and you just want everyone to be safe until dismissal. Learning would be a MAJOR victory but it seems like a far fetched dream... 

Or is it? 

I save my best trick for the day before breaks. Math scavenger hunts :) Ahead of time, I prepare, essentially, task cards and tape them up around the school. Students are broken into teams and they go off and do math review. But because they are in a team and walking around the school it seems super novel and they have no problem at all doing a set of math problems quickly and accurately! 

I would recommend making the tracking sheet novel in some way. At Thanksgiving, I have students collect a turkey feather at each station, when their turkey is complete with all 8 feathers, they are done. At Christmas, they are collecting ornaments as they go. When their tree is fully decorated the hunt is over. Before April break? Easter egg hunt with fake money inside. Get counting kiddos :) 

Also, if your students tend to bring in a CRAZY amount of snacks on the day leading up to a break, it is fun to set up a snack in the classroom and allow students to come back at the 1/2 way point for a special treat. 

I am telling you, I have used this strategy for years and it works. every. time. 
Holiday Gift for YOU!
Want to try the scavenger hunt? I put together a scavenger hunt for you! I have included one set of task cards appropriate for grades 1 and another appropriate for grade 2. Both use the same game board. Click the collage below to grab it. Enjoy! 



Thank you so much for stopping by today! Tomorrow, head over to Math Coach's Corner for the next day of the iHeart Math Holiday Hop!
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Saturday, December 6, 2014

Keywords are NOT the Key to Word Problems

Solving word problems is hard.

Really, really hard.

In a world where reading comprehension, logical thinking, math computation and visualization come together, word problems were born.

As teachers, we are interested in doing everything we can to make instruction make sense, come alive and "click" for our students. And, unfortunately, in the interest of the end goal it can be very easy to try to teach tricks to our students. In the long run, however, this does SUCH a disservice!

Let's look at the list of commonly taught "keywords" and match it up to a progression of addition and subtraction word problems from K-2 to see if learning these key words will serve a student well.

Keywords for addition often include:
add                      
sum
total
plus
and
in all
altogether
together
more

Subtraction often sounds like:
difference
take away
minus
fewer
less
took
gave away
left over
difference

First up: Do the keywords hold up in Kindergarten? 

CCSS.MATH.CONTENT.K.OA.A.2
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.


So, the standards alone here really aren't enough to determine what word problems look like at this level. To learn about what the standards "look like" you need to look into the progressions document. Have you seen this chart? I know it's difficult to read, so I have included a link.
Commoncoretools.wordpress.com
Basically, the chart lays out the different type of word problem and then, through the shading, explains which problem types are expected at each grade level.

Back to kindergarten now. The bulk of the problems are put together, take apart, add to and take from problems where the result is the missing piece. Looking at the questions, one by one:

4 bunnies sat on the grass. 5 more bunnies hopped over. How many bunnies are on the grass now?
Keyword indicates addition, addition of 4 + 5 will solve the problem. 

10 apples were sitting on the table. I ate 4. How many are on the table now?
No keyword... you could argue that "ate" means take away so that's subtraction but, in the world of keywords they really offer no help here. 

2 green apples and 4 red apples are sitting on the table. How many apples are on the table?
Keyword indicated addition, addition of 2 + 4 will solve the problem. 

Grandma has 10 apples. How many can she put in the red vase and how many can she put in the blue vase?
This question is open ended and contains no keywords. 

Conclusion? In kindergarten, keywords are not misleading, however, they are not helpful in solving all types of word problems.

Next up: Do the keywords hold up in 1st Grade?

CCSS.MATH.CONTENT.1.OA.A.1
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1


The "1" at the end of this standard is directing you to the chart that describes the problem types. Too bad they didn't make the same note in the Kindergarten standard.... I digress...

So here you see we have unknowns in all positions in all types of word problems. The only real caveat here is that change problems where the start is missing and comparison problems where the language is intentionally misleading are saved for second.

Let's go through a few problems:

6 bunnies were sitting on the grass.Some more bunnies hopped there.Then there were 12 bunnies. How many bunnies hopped over to the first 6 bunnies?
Keyword indicates addition. 6 + 12 does NOT solve the problem. A student needs to be quite flexible with situation and solution equations for this key word to make sense. 

18 apples were on the table. I ate some apples. Then there were 10 apples. How many apples did I eat?
No keyword. If you were in the camp that said "ate" indicated subtraction before 18-10 will yield the correct answer. 

14 apples are on the table. 7 are red and the rest are green. How many apples are green?
No keyword.

Lucy has 16 apples. Julie has 9 apples. How many more apples does Julie have than Lucy?
Keyword indicates addition. Addition will NOT solve this problem. Without any obvious action in this word problem, even a situation equation is a difficult argument for a 6 year old. 

Lucy has 10 fewer apples than Julie. Julie has 18 apples. How many apples does Lucy have?
Key word indicates subtraction. Subtraction will solve the problem... but wait until 2nd grade... 

Conclusion? In first grade, many problem do not have obvious keywords and those with keywords may be very misleading with a keyword leading to a situation equation rather than a solution equation. But when was the last time that you saw a "keywords" poster that said "Keywords for determining the operation in your situation equation!" I haven't seen one yet. 

If I haven't yet convinced you that teaching keywords is doing a disservice to students, continue on to reading about 2nd grade. 

Last, but CERTAINLY not least: Do the keywords hold up in 2nd Grade?

CCSS.MATH.CONTENT.2.OA.A.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

I am not going to even TOUCH the multi-step nature of word problems at second grade. Obviously with multiple steps there may be multiple keywords for a student to sift through.

Here we add in the last 4 type of word problems.

Some bunnies were sitting on the grass. 19 more bunnies hopped there. Then there were 27 bunnies. How many bunnies were on the grass before?
Keyword "more" indicates addition. 19 + 27 will not solve the problem. Addition is appropriate only in a situation equation. 

Some apples were on the table. I ate 7 apples. Then there were 45 apples. How many apples were on the table before?
No keyword unless we count "ate". In that case, subtraction is appropriate. 

Lucy has 19 fewer apples than Julie. Lucy has 36 apples. How many apples does Julie have?
Keyword indicates subtraction. Subtraction will NOT solve this problem. In fact, a solution equation using subtraction would have to look like J-19=36. How many of your students would have written that? In terms of pure key words most students would write 36 - 19 as Lucy was written directly before the word "fewer". The keyword is very misleading in this problem! 

Julie has 63 more apples than Lucy. Julie has 89 apples. How many apples does Lucy have?
 Keyword indicates addition. Addition will NOT solve this problem. Most students, in an attempt to write a situation equation matching addition will write 89 + 63 = L as Julie's name was written first. The keyword, again, is very misleading. 

In fact, in the problem type diagram for the last 2 problems, the descriptions are "fewer suggests wrong operation" and "more suggests wrong operation".

Overall conclusion: 
In kindergarten keywords don't always help but they don't really hurt either. However, if students in K are taught key words, they are set up for trouble in first grade where key words lead to situation equations but NOT necessarily the answer of the problem. By third grade, key words can be misleading in a way which will cause students to write even a situation equation in the wrong way.

Suggestion? Focus on the action of the problem, what is a part, what is a whole. Are we missing a part? Use subtraction or missing addend addition. Missing the whole? Use addition. In comparison problems, focus on the larger amount as the whole, and the parts being the smaller number and the "more/fewer" piece.

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Thursday, December 4, 2014

Fluency Through Flexibility (Book Review)

A few weeks ago "Fluency through Flexibility: How to Build Number Sense" showed up on my doorstep. I devoured the introduction and was immediately taken with the straight forward, research based, easy to follow nature of the text. I was hooked.

So often, when reading a professional text, I agree and understand the premise of the book but struggle with merging that information with the demands of district curriculum or a given "program". Sound familiar? This text built my knowledge around building number sense and then provided a variety of activities I could implement tomorrow through quick activities for small groups or the whole class. If you teach intervention groups, these activities are PERFECT for RtI.


To give you some background around the basic purpose for the book; The author states: 
"This book is built on the premise that if we spend time in the early grades letting children explore and develop their informal knowledge, it will build a child's number sense and increase their flexibility when working with numbers, then fluency will actually occur faster. Plus, by developing a sense of relationship between number, children will be able to transfer that number sense when working with multi-digit numbers as well. 

Love it. Get it. But, what next? How do we "Develop a sense of relationships between numbers" in an organized, research based effective way? The book outlines the 4 relationships which are most beneficial to student's number sense development. 
  • Spatial Relationships
  • One/Two More and Less
  • Benchmarks of 5 and 10
  • Part-Part-Whole
Following a thorough discussion of each relationship, it's importance in building number sense, and how an understanding of this relationship will allow students to form a foundation for further math understanding, the author provides 15 activities that you can do in order to promote student's knowledge of this relationship. 



The book is in black and white. I am quite the active reader :)
 Each activity is laid out with a quick description of materials, steps, what to watch for in your students and ideas for reusing the activity in the future at a higher level or in a different way.

One last feature I appreciate about this book is the interview style assessments provided in the appendix of the book. A separate interview has been included for each section of the book.

This text has helped to guide me in my math RtI first grade groups in a number of ways. First, I feel confident in using the activities because they are research based. The book is littered with references and I know that if an administrator walked into my math group that I would be able to justify the work with my RtI students. Also, in structuring a 30 minute group, I am able to choose activities from the 4 sections of the book that work together and build upon one another. For example, in a group earlier this week, I did the Story Time activity from the spatial relationships section of the book. The activity focused on representing numbers in a counting story. While students were focused on representing numbers on a ten frame, the activity also touched on the relationship of "one more". Following the story, we capitalized on the "one more" nature of the previous activity to move into an activity from the One/Two More & Less section called Ten Train where students rolled a +1/+2 dice to build a tower from 1 to 10. Planning a research based lesson that builds in a logical manner was VERY easy using this text.

If you are interested in looking into the text for yourself, it is available on the website "Mathematically Minded" which is linked in the picture of the cover above as well as HERE.


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