Over the next few blogs, I will be interviewing Barry Garelick and J.R. Wilson about their new book, Traditional Math: An Effective Strategy that Teachers Feel Guilty Using. You can read Part 1 and the rest of the saga as it comes out, here.
Q2: I can picture students becoming very fluent with the procedures, and yes, the concepts, when they experience a traditional math consisting of explicit instruction. Can you explain how traditional math deals with mathematical thinking and reasoning?
JR: Mathematical reasoning is important and has been heavily emphasized in recent decades. Unfortunately, reasoning is not well defined for teachers, if at all, and little to no guidance is provided on how to teach (develop) or assess student reasoning. It is not like teaching and assessing student ability to do subtraction or multiplication. While reasoning is only briefly addressed in our book, it is developed throughout the book.
Descriptors or definitions of mathematical reasoning vary. NCTM’s 1989 Curriculum and Evaluation Standards for School Mathematics says, “Mathematics is reasoning.” NCTM’S 2000 Principles and Standards for School Mathematics says, “Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts.” The Washington Exemplary Mathematics Standards: 2008 (WEMS) includes reasoning in the process strands saying, “Reason mathematically. Be comfortable gathering data, analyzing evidence, and building arguments to support or refute hypotheses.”
The section of the book that I authored focused on the fundamental skills and concepts students need for problem solving which requires math reasoning. I did not address solving story problems even though elementary level instruction should always include such work.
Research in cognitive science shows that memorized facts and procedures are important to the development of math reasoning. Such research provides evidence and a growing body of knowledge supporting the Science of Math. The strategies and approach in our book are supported by and in line with the Science of Math and the findings of cognitive science.
With little mention of math reasoning in our book, reasoning is developed all along. Math fact fluency is developed. To be clear on that, facts need to be memorized. In the book, I say, “Quick and accurate recall of basic math facts is math fact fluency.” Also, efficient and workable algorithms are developed. Throughout the book, students are asked general and specific questions related to what they are working on. These questions help develop math reasoning. Here are examples of some general questions:
- What do we do next?
- Why would we do that?
- Can anyone explain why doing that works?
- What happens when you…?
- Do you see a quick way to work the problems based on what you understand?
- Who can give an example of this?
Multi-step problems give students practice applying their developed reasoning skills. The same kind of problems may be used to assess math reasoning. A teacher can create their own story problems that employ recently taught skills. Here are some example questions from the WEMS that are the kind that might be used to practice the application of or assess mathematical reasoning.
- Grade 2
- After adding 5 apples to a bag, Diana had 12 apples all together. How many apples were in the bag initially?
- Grade 3
- Maddie, Jose, and Maris divided their lemonade stand profits equally. If they made a total of $9.36, how much money did each one receive?
- Lyng had $3.45. She bought 2 pencils for 75¢ each. How much money did she have left?
- Grade 4
- Elliot got $40 for his birthday. He spent 3/8 of the money on a video game. How much did the video game cost?
- Shalimar bought 5 lbs of grapes. She gave the cashier $20 and received $5 in change. Find the cost of 1 lb of grapes.
- Grade 5
- Dawson purchased a box of 24 oranges for $4.50, and sold all the oranges in packages of 3 for $1.05 per package. How much profit did he make?
- Grade 6
- Niki had $50. She went shopping and spent 60% of her money on clothes and 60% of the remaining money on food. How much more did she spend on clothes than food?
- A glacier moves about 5 inches every 8 hours. Write the ratio of distance to time and use it to find how far the glacier will travel in 72 hours.
Barry: I would add that procedures themselves help develop mathematical reasoning. Generally, the thinking/reasoning is an extension of learning and mastering certain problem-solving procedures. For example, students learn what a fraction of a whole number is, like 1/3 of 60, by seeing it as a division problem. That is, it’s the same as 60 divided by 3 which can be represented visually in a diagram. They have also learned about fraction multiplication, so they can be shown that 2/3 is the same as 2 x 1/3. These principles can then be extended to find 2/3 of 60. If 1/3 of 60 is 60 divided by 3, or 20, then 2/3 of 60 is 2 x 1/3 of 60, or 2 x 20 or 40. The procedure leads to reasoning by seeing the pattern. The patterns are a form of reasoning and mathematical thinking.
I also agree that multi-step problems promote mathematical reasoning. JR provided some examples of this. Such problems when scaffolded, can build to allow students to solve a more complex problem. In the warm-up problems that I have students work on at the beginning of class, I will sometimes include a word problem that poses a challenge and which is built upon in succeeding days. For example, the following problem will be presented in the warm-ups: “The average of two numbers is 23. What is the total of the two numbers?”
This can be solved algebraically, but also numerically. Let’s look at the numerical approach, because I usually pose this in seventh grade. Students will be stumped initially, but have had some practice in “working backwards” to solve some problems, by reversing operations. One prompt I give is “How do we find the average of two numbers?” Hearing that the two numbers are added and then divided by two, I’ll ask “How do we work backwards?” With some thought they see that multiplying by two will give the total, which provides the answer to the problem.
The problems get more complex on succeeding days. For example they may next see a problem like this: next day, the warm-ups may contain a problem like this: “Tim’s average score on the last four math tests is 95. A) If on his fifth test his average is 96, what is the total of his five scores? B) What was his score on the fifth test?”
Students know how to solve Part A from previous problems. But, if a student is stumped I may say: “We’ve solved similar problems. Remember working backwards?” Now the reasoning comes in. They recall that for an average of two numbers, multiplying by two yielded the total. So now, if they multiply by five, they will get the total of those five tests. This total is then used to answer Part B. A prompt might be “Do we know what the total is for the four tests? Look at the problem again.” The first sentence provides the clue. The average of the four tests is 95, so multiplying by four will give us the total. Now we have two totals: The first four tests, and then all five tests. Students will then reason that the difference of the totals will provide the score for the fifth test.
The next day I may provide a problem that is solved in the same way, but is worded differently: “Maria has averaged 88 on her last three tests. What does she need to get on her next test to obtain an overall average of 90?”
For students who are stuck, I will ask how we solved the problem about averages the day before. Using reasoning some students will see that finding the difference of the two totals (i.e., total score of four tests minus total score of three tests) provides the answer to this problem as it did the last one. Those who do not solve it, will usually react in the following manner when they see the solution: “Oh, I should have got that one.” This tells me that they know what procedures and steps were involved and how they were used to reason through to a solution.
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