Part 2 of ‘using strategies based on properties of operations’…

In my last post I assigned two problems for you to solve mentally and then think about the properties being used in your strategy.

Compensation or Friendly Numbers        58 + 36

To solve this problem mentally using the strategy of compensation, I chose to make one of the given addends into a friendly number to work with in my head. I know that I can add 60 in my head much easier than I can a 58. So I’m going to move 2 from the 36 over to the 58 and now my problem is 60 + 34 which is 94. Easy, huh?

Now let’s investigate why this works. First I have 58 + 36 = 58 + (2 + 34). I have simply decomposed 36 into 2 and 34, knowing that I need the 2 in order to make a friendly number. Now using the associative property of addition, I add the 2 to the 58 instead of the 34. (58 + 2) + 34 = 60 + 34 = 94.

Breaking apart into Place Value                 158 + 221

I probably don’t do this problem exactly as the strategy calls for, because to me it makes more sense to not break apart the first number. I know that the answer is 379. How? In my head I look at 221 = 200 + 20 + 1. Then I add the 200 to the 158 and get 358. Next I add the 20 to 358 and get 378 and finally I add 1 to 378 to get 379.

What properties of addition did I use? I’m not even really sure. I guess it could be called the associative property of addition. 158 + 221 = {[(158 + 200) + 20] + 1} = 379. If you have any other idea(s) please share them. I really just used what I knew about place value to solve this problem. If you look back at those Common Core mathematics standards, you’ll notice that students are supposed to solve problems based on their understanding of place value and the properties…

Let’s look at some of the subtraction strategies next. So your next assignment is the following:

335 – 219                             413-135

Remember to see if you can determine what mathematical properties, place value understandings, or relational understandings you are using to mentally solve these problems. Go back and try to solve the problems a second time using a different mental strategy. I’ll be back in a couple of days.

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For Teachers’ Eyes Only

Throughout the Common Core Georgia Performance Standards for Mathematics in K-5 you read the phrase “based on place value, properties of operations, and/or the relationship between addition and subtraction; as well as multiplication and division”. What does this mean?

Let me provide a context to which we can all relate. Most all of us were taught “THE” properties in one to two days each year we were in school. There was one page in the book that described each property – commutative, associative, distributive, identity – and provided one to two examples of each. Then on the next page in the book we were given numerous examples and then we had to record what property was being presented in each example. Not only was it boring, but rarely did we see a purpose.

So when I taught 6th grade I decided to make it more ‘exciting’. I typed up a list of examples, cut them apart, handed one out to each student then had them walk to the designated part of the room for that property. Yes, students may have enjoyed it a bit more, but they did not really make any connections between what they were ‘matching’ and the other problems they did in mathematics the other 178 days of the school year.

I recently taught a workshop on how Number Talks help students develop understanding of these properties. I stumbled a bit on the first time of presenting this idea, but improved with practice and additional thought.

Doubles/Near Doubles    999+999

I honestly do not know what 999+999 is without thinking about it. But I can determine it rather easily. 999 is really close to 1000 and I automatically know that 1000 + 1000 is 2000. So now I know that 999 + 999 is 1,998. Wait a minute, how did that happen?

999 + 999 = 999 + (1 – 1) + 999 + (1 – 1)  Behind the scenes I added a ‘zero’ to each 999         without changing the value of the problem. This can be done because zero is the Identity Element of Addition. Zero does not change the identity of any number that it is added to.

999 + 999 = (999 + 1) – 1 + (999 + 1) – 1 This is where I said, “I know that 1000 and 1000 is 2000.” But what went on behind the scenes? After I added the 0 to each 999 in the form of (1-1) I then moved my parenthesis to add in a different order, using the associative property of addition.

999 + 999 = 1000 -1 + 1000 – 1 = 1000 + 1000 – (1 + 1) There are two properties that I used to write this equation string. I swapped the first -1 with the second 1000 using what is called the commutative property of addition. Then I should have been left with -1 -1. Where did            – (1 + 1) come from? I could have thought in my head “2000 minus 1 is 1999 and then minus 1 again is 1998.” But I actually thought in my head “2000 minus 2 is 1998”. I was using the distributive property of multiplication over addition. If you can remember back to algebra the negative is to be distributed to each term within the parenthesis – ( 1 + 1 ) = -1 -1.

In this one simple problem I used 4 mathematical properties ‘behind the scenes’. Why do I keep putting it like that? Teachers need to know what is ‘allowed’ and ‘not allowed’ so that they can assess on the spot whether a student is using a mathematically valid strategy. Yet, the students do not necessarily need to know the names of these properties depending on their level of understanding.

So when a student talks through the problem like I did originally, they have in fact used the CCGPS for their grade level that says students should use strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division. I have discovered that the difficulty associated with these particular standards is that teachers may not always recognize unspoken properties. I hope that this series of conversations will be helpful.

I stated that I do not believe that students need to know the names of these properties; however, when a student changes a problem by switching the placement of two numbers the teacher should ask why they did that and does it change the value of the problem, why or why not? After this ‘switching of numbers’ becomes so commonplace in the classroom then the teacher can introduce the official name of commutative property of addition (or multiplication). I know that tacking on ‘of addition’ or ‘of multiplication’ is time consuming but if you consistently use the whole name of the property once it’s introduced at some point one of your students is going to ask, “Is there a commutative property of subtraction?” What a beautiful question! Respond with, “That’s an excellent question. Talk with your neighbor about this question and see if you can determine if there is such a property and be able to explain why and why not?”

I will continue to discuss some of the Number Talk properties in following posts. But in the meantime, you have homework.

Solve the following problems mentally and record your thinking. Then go back and look at your recorded thinking and see if you can determine how you used strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division.

a. 58 + 36     b. 158 + 221    

I’ll assign some subtraction problems next time.

Timed Tests – Are they Helpful?

Before I get into how I really feel about timed tests, I must share my confessions. I started teaching 6th grade mathematics in the early 1990’s. My students did not know their multiplication facts. (I hear this comment still today from 3rd-12th grade teachers!) So I gave a timed multiplication test every Thursday. There were 34 problems on the test and each one was worth 3 points. So if you answered all of them correctly, you could earn a 102. The students would typically have anywhere from 2 to 3 minutes. I never told them how long I actually gave them, they were just told that they had 2 minutes. I did NOTHING between each Thursday to help my students develop an understanding of multiplication either. (It was not in my standards.) So week after week I had the same students scoring 102 and couldn’t wait for the next Thursday. I had another group of students scoring a 42 every week and they acted like they could have cared less about the test. Then there was the third group of students scoring between 78 and 90 each week. They walked into class each Thursday with sweaty hands and some even sick on their stomachs. And for the most part, no one moved out of the range of scores I just provided.

I began teaching in January that year and replaced a teacher who retired in December. She had the reputation of being a great disciplinarian. For you teachers reading this, you know what that means…she had all of the ‘hard to manage’ kids on her team. That may have been a good plan in the past, but placing a brand new teacher in the same position was not very kind. I was either writing students up or writing assignments for them while they were in In School Suspension (ISS). Our team kept getting notices from the ISS teacher stating that we weren’t sending enough to keep them busy. So if they had any ‘free’ time, the ISS teacher just assigned a page or two out of the dictionary for the students to copy. Ugh! Are you kidding me? So I decided if the students were going to copy something it would be something that would benefit me. : ) I began sending a sheet of written out multiplication facts and had my students copy that during their ‘free’ time.

It didn’t take long to discover that those students started scoring higher on their multiplication tests each Thursday. About that same time the parents of the ’42 group’ were calling and complaining. So I came up with an idea. For every complete set of multiplication facts a student turned in, he/she would earn up to 10 points to be added to their lowest multiplication test (a set consisted of the twos-twelves). No one could raise their score above a 100. That was reserved for those who correctly answered all 34 problems on the first try. Yes, scores began to rise dramatically.

I’ve read research that supports timed tests as long as you are doing something to help the students improve in between the tests. (I have an idea of the source of that information, but I’m afraid to put it here without being certain.) The scores should not ‘hurt’ the students’ grades and the student should be encouraged to better their score, not necessarily to score a 100. I’ve also read research that says if you hear, say, do and write something you are way more likely to remember it. So all I had done was add the ‘write’ component.

Although I experienced success with my methods at that time, I now question my motives. My main goal was for my students to learn their multiplication facts – and most of them did. But the students who had scores improve on the timed multiplication test did not improve their scores on anything else they completed for me. I was not teaching for understanding. I now realize that my motives were focused on short term goals and not what was best for the students in the long run.

Since my early years of teaching I’ve read more research – the kind that goes against timed test. http://joboaler.com/timed-tests-and-the-development-of-math-anxiety/ is a wonderful example of this and she even provides links to the research that supports her article. Timed tests can (and does in many cases) create or contribute to a hatred of mathematics. It took a while for this to sink in to my thick head because ‘I had seen the success in my own classroom’. Once I began thinking about the logic, my own beliefs started to change.

Timed ‘test’ does not have to exclusively mean a test; it could be the use of flashcards, or the class game ‘Around the World’. Consider the kids who already know their facts. Are they learning anything by using flash cards, playing “Around the World” or taking a timed test? This is just a waste of time for them.

Now consider the kids who simply do not know their facts. What will the timed tests, flash cards and playing ‘Around the World’ do for them? My guess is that they will act out, become angry, develop greater self helplessness in mathematics, and learn to ‘not care’. I’ve seen this happen. All of those outcomes are the only way many of those students know how to cope with disappointment, embarrassment, and frustration.

Next consider the kids who are middle of the road when it comes to knowing their facts. They know the basics but flounder when it comes to the 6, 7, 8’s. The ones they may know this week they don’t know next week and vice versa. They’re the ones who get all worked up over the timed tests, because right now they still care. During ‘Around the World’ they aren’t hearing the facts that are given by the other students, they are just sitting there praying for the teacher to call out an easy fact when it gets to them; or better yet, they’re praying for a fire drill or for class time to run out. Some of these students may learn more math facts as a result of the negative pressure, but others just begin to move into the frustrated, angry, depressed, “I don’t care anymore” category.

So let’s recap the scenario above regarding timed tests, flashcards, and ‘Around the World’. Who benefited? Only the few in the middle group who actually used the negative pressure to cause them to try harder. For everyone else it was a waste of time and more specifically even detrimental to many.

So am I saying that students should not learn their math facts? Of course not! I know that life mathematically is a whole lot easier if you know your addition and multiplication facts. But students should be learning their facts with understanding. For example, they can learn their facts through the use of Number Talks and Cognitively Guided Instruction.

Did I open a can of worms today?

Number Talks

Number Talks

Mathematical discussions have taken place in many classrooms for many years. However, the book titled “Number Talks: Helping Children Build Mental Math and Computation Strategies” by Sherry Parish has greatly influenced my thinking. Number Talks is a specific time set aside within a classroom when students mentally solve problems and then share their strategies with their peers. The book itself provides ideas on how to get started and also includes a DVD that shows Number Talks at several different grade levels. In the book the teacher can also read about different strategies students may use when solving problems mentally. Some of those strategies are Make a Ten, Friendly Numbers, Compensation, Near Doubles, etc. Special strings of problems are also provided in the book. A string is a set of problems used to highlight the relationships between problems. Pages of strings are provided for certain grade bands. However, if your students are in an upper elementary grade and Number Talks are new to them, they may not be ready for the strings provided for that grade band. It may be necessary to use some of the K-2 strings first. As a teacher you may decide not to use all of the strings for a particular strategy but I would recommend that you not randomly choose strings. Be intentional about the strategy strings that you choose. If you want more information or clarity please feel free to ask. There are a lot of resources regarding Number Talks on the internet. http://schoolwires.henry.k12.ga.us/domain/3791 is one of those sites. I have been told that there are many YouTube videos available also.

I have been asked several questions regarding number talks and would like to address them below.

  • Can I teach the strategy before I give a string so that the students know what to do? I say ‘no’. If after several days of working on similar strings no one in your class has shared the desired strategy, then I think it would be appropriate for you to share a strategy that you “saw a student use once”. You may wonder how this is different. If you share a strategy first then the students will be more likely to try to follow your lead and not try to solve the problem with a strategy that makes the most sense to them.
  • Is it Ok to tell the students what strategy to use? When a student shares a strategy I do believe that it is Ok for you to ask the class to try to solve the next problem using “Marcia’s strategy”. But I would not require that students use a particular strategy all of the time.
  • My students try to come up with the most obscure way of solving a problem, what do I do? Many students simply want to share their ideas with others. However, you want to give everyone an opportunity to participate during Number Talks at least once a week. Some of these students can be satisfied by simply sharing their strategy with an ‘elbow partner’. Before you call on someone to share in front of the class, tell the students to discuss their solution and strategy with their ‘elbow partner’. This way everyone can talk.
  • Don’t we want our students to be efficient? Yes! But you can’t force a child there until he/she is mathematically ready. So in the meantime, spend time asking them to compare strategies used to solve the same problem. Then talk about efficiency.
  • The next is not a question but something that a teacher shared with me this past week. During a curriculum night with parents, she conducted a Number Talk. She said that parents were amazed and very excited to know that their children were talking about mathematics in such a way. I think this was a great idea, so I suggest that you try it sometime.

This is a question for you? Does anyone know of a rubric for assessing Number Talks? Teachers are looking for a way to periodically ‘give a grade’ for student understanding displayed through Number Talks. If you have such a tool will you please share it with us? Thank you.

Well, that’s all for this post right now. I’ll address any additional comments or questions in further posts.