Cognitively Guided Instruction

Cognitively guided instruction is a very fancy term for a simple, yet incredibly effective teaching practice. I read students word problems (using a variety of types and structures), students solve them on white boards or paper using unifix cubes, pictures, number sentences...(depending on what strategy they are comfortable solving the problem with), several students share their strategy, students solve the next problem with, hopefully, a more sophisticated strategy than the one they were using before after having been inspired by the students that just shared their strategies. Obviously, students learn at different rates, and it may take some students seeing a strategy several times before they feel comfortable enough to use it.

Benefits for me: I get to see strategies students are using like a window into their brain as well as see the growth over time on the strategies they are using. I really experience joy when students use sophisticated strategies that are different than I thought of (almost daily).

Benefits for students: They get to experience the idea that there is a correct answer, but how we get there may be different because we all have different brains and that all strategies are valued. Most of all students learn that math is more than learning a set procedure. It involves understanding the operation and choosing the best strategy for the student and for the specific problem. They get to see and "borrow" strategies from their peers and are always moving to more sophisticated strategies.

A side note: Students love when I name strategies after them. I record strategies on a chart. Then, minutes or weeks later I may say "Oh, you used the Ophelia strategy, so clever." or "Why don't you try the Ryan strategy for that problem?" It really leads to the feeling of being a serious mathematician and making discoveries about math.

example- 8 kids got on the bus, then 7 more got on the bus. How many kids are on the bus now?

Students are always moving at their own pace to more sophisticated strategies like directly modeling all the parts of the problem (taking out 7 manipulatives, then 8 manipulatives, then counting all of them. Counting up or counting from students just think or say "8", then count up 7 more on their fingers or in their head. Derived fact- students use a fact the know to find a fact they don't know (I know 8+8=16, so 8+7=15).

I can make a problem more difficult for the class by changing the numbers or using another problem type. (8 kids were on the bus, then some more got on at the next stop and there were a total of 15 students on the bus. How many students got on the bus?)

Here are some links for CGI:

Cognitively Guided Instruction book

Great set of CGI Problems to use- written by Mary Alice Hatchett

Subtraction Fact Strategies

Here is our chart of subtraction fact strategies. Please forgive my terrible handwriting. I have never mastered that perfect teacher handwriting. Again, I don't require students to use these strategies, but I do want to have some common names for strategies we can all refer to and some strategies to suggest when a student doesn't have a strategy in mind to use. My favorite strategy on this list is the "up to ten" strategy. It is a counting up strategy from the lower number. We ask ourselves how many up to ten, then how many after ten and add those two partial differences together to find the difference. This is great to demonstrate on a ten frame and/or number line. For example, for 15-8 I would think 8 + 2= 10, and 10+ 5=15, and 2 + 5=7, so 15-8=7. This is a powerful strategy because it helps kids with the facts they find most difficult and because it is transferable to larger numbers (25-18=7).

Some strategy names are from Bridges math program

Break Apart game

I will admit that when I saw Greg Tang speak years ago at a math conference I got a little math crush on him that has never gone away. When he talks about his beliefs and ideas about how math should be taught, I find myself nodding in agreement like a bobble head doll. I love this game, Break Apart, on his math games site. It has students drilling fast strategies instead of just drilling facts they don't have a strategy for. It is worth the small subscription fee.

This game would be great for individual student practice as well as whole group on an interactive whiteboard.

Addition Strategies

At the beginning of the year in third grade we review addition strategies. My goal is not to make students use a particular strategy, but to give some common language to efficient strategies so that, in the future, we can discuss them using terminology we all know. I pose an addition problem, students solve, and I have several students share their strategies with the class. I tell them that there are many fast, smart strategies to use, but that they can use the strategies on this chart when they need to. I refer back to it when I am helping a student that is using a slow or ineffective strategy for the next few months.

Addition Strategies- many names are from Bridges math curriculum

I heard somewhere, I don't remember where, that we don't really memorize many addition facts even as adults. We just do a fast mental strategy. I doubted that at first, but I think it is true. Think about 8 + 6, for example. I don't have it memorized, but I think 8 + 2 = 10, then I add the 4 left in the 6 to =14. I do all that in less than 2 seconds. So repeating or drilling the fact over and over without having a fast strategy in place first may be a waste of my students' time.

Addition and Subtraction Strategy debate

I somehow stumbled on this old article explaining multiple digit addition and subtraction strategies.

What Happened to Borrowing and Carrying?

What is interesting to me is not the article, sorry author, but the debate in the comments thread that ensues. I have often had trouble understanding why a parent or teacher would be opposed to their child knowing and understanding more than one strategy for addition and subtraction. The comments on this thread kind of enlightened me. Some people view these additional strategies as unsophisticated or stepping stones to the traditional algorithm,  slower than traditional methods, or just don't realize the power of students building understanding rather than just memorizing a procedure. One parent that commented was upset that her child was getting poor grades in math and understood the traditional algorithms, but not these new ways of adding and subtracting. I got her point, but still thought that she didn't understand the bigger issue here. Student created and alternative algorithms help students understand the concepts at a deeper level and help connect ideas in math. I always have students that have been taught to borrow and carry by their parents at home. I love that these parents want to do math at home. The difficulty is when students or their parents think that these strategies are the ONLY or BEST strategies.

I tell students in my room that they can use any strategy if it fits these guidlines...

1. You must understand the strategy and why it works.
2. It must lead the the right answer for you the vast majority of the time.
3. It must be efficient. 

One of two things happen to students that have parents who taught them the traditional algorithms. They either, one, abandon them because they don't understand why it works or find a strategy they understand better or, two, learn why their strategy works (Why do we write a one there or change that number to a 12?). Either way I am a very happy math teacher.

My journey teaching math problem solving

When I started teaching I was working as a Title I Math Specialist 10 years ago pulling whole classes K-4 to teach math problem solving. Problem solving was viewed as a completely separate part of math, possibly even a separate subject, not connected to what students were learning during math time in the classroom. I was expected to teach a specific series of steps to solve problems. Students had to read the problem two times, underline important information (and sometimes write a bulleted list), write a sentence starting with "the problem I am trying to solve is...", solve the problem using the strategy being taught (work backwards, make a list...), put a box around the answer, write a paragraph about how they solved the problem, and some other steps I am surely forgetting.

While I have always loved teaching problem solving in math something about teaching it that way always seemed phony and I had a suspicion that they students thought so too. Many of the things I made students do were assuming they were not capable of solving the problem (making a list of the important information, writing the sentence about what they were trying to solve, and using a prescribed strategy), made to do to make the teacher's job easier (putting a box around the answer and writing a paragraph to explain thinking), and just not authentic to the the way we problem solve in real life. There were good lessons for students in there being wrapped in bad packaging.

I was in that position for 2 1/2 years. Now I am teaching 3rd grade (after teacher 4th grade for a few years) and have come to the conclusion that...

1. Problem solving is probably the most important set of skills I can teach students in math.
2. Students are innately good problem solvers and do not need me to tell them what strategy to use or a step by step procedure for problem solving.
3. Problem solving is not separate from the rest of my math curriculum and can and should be woven into my daily instruction.
4. Students develop math skills and become more sophisticated problem solvers by hearing other students share a variety of strategies and their thinking and through minilessons taught by the teacher.

I have a picture in mind of what I want my math classroom to look like incorporating those beliefs as well as others I have about teaching math. The picture is clearer on some days than others. I am using this blog as a way for my to sort out my thinking, record what is working and what is not, document my journey as a math teacher, and to collaborate with other teachers.