If one was to categorize every SAT question from the very first SAT ever published, I believe we would find the following type of *algebraic ratio problem *one of the most common type. Even with all the exposure students now have to SAT problems, my direct experience is that many students still struggle with these types of questions.

**WHY?**

More importantly, are these types of problems important enough in the CCSSM to justify the time investment to introduce them in middle school and reinforce in secondary algebra classes? IMO, ABSOLUTELY!

**If h hens consume a total of p pounds of feed per day, then, at this rate, how many pounds of feed would c hens consume in x days?**

Not only was a similar question the recent *SAT Question of the Day* on the College Board web site, the statistics were also published:

*35620 responded (up to the time I checked)*

*31% correct*

*So, about 7 out of 10 students attempting this question online got it wrong.*

Note: The actual question was followed by 5 choices, allowing students to plug in numbers and test each choice, but I chose to focus on the question here rather than test-taking strategies.

IMO, the College Board hires highly competent math people who write succinct, accurate and helpful online solutions but this only scratches the surface. It only suggests one particular approach and has little to do with Instructional Strategies and the various ways children develop these important ideas.

**REFLECTIONS...**

1. Where are ratio concepts introduced for the first time in the CCSSM? K? 1st? 4th 5th?

2. By your own estimate, how many of these kinds of questions appear as sample problems or homework exercises in your elementary/prealgebra/algebra texts?

3. Do you believe ALL your students receive adequate exposure to and review of these?

4. Would you be willing to share some of your favorite methods of laying the groundwork for and developing the skills and concepts needed for your students to be successful with ratio problems and ultimately algebraic types? If I take a risk, would you?

Putting myself out there...

The simplest and most instinctive approach usually makes the most sense, doesn't it? We know how *we* learn best and the same is true of all students. Do you accept the following as a truism, an essential tenet of teaching and learning mathematics?

**EVERYONE LEARNS BETTER WHEN PRESENTED WITH CONCRETE NUMERICAL RELATIONSHIPS BEFORE TACKLING ABSTRACTIONS. FURTHER, THE COMPLEXITY OF LANGUAGE SHOULD BE GRADUALLY INCREASED, STARTING WITH THE MOST ACCESSIBLE INFORMAL PHRASES.**

**For example,**

*If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would one hen eat in one day?*

When first introduced, should our focus be on which operation to perform? In my view, **our goal should be to develop number sense, in this case, ratio sense. **

**We all know that a powerful construct for developing**

*ratio/proportion sense*is the idea of first reducing the information to a UNIT.Many of us were taught this way and most children tend to think like this at first.

Scaffolding...

*If 6 hens eat a total of 12 pounds of feed each day, how many pounds of feed would nine hens eat in one day?*Working from

*one hen consumes 2 pounds per day*, the child can usually move on to 9 hens eat 9x2 or 18 pounds per day.

Two points here...

First, I believe it is important to routinely use a variety of equivalent phrases:

"

*in one day*" vs. "

*each day*" vs. "

*per day*."

Secondly, I would encourage students who can reason proportionally to share this with the group:

**"**

*Well, if 6 hens eat 12 pounds, then 3 hens will eat half as much or 6 pounds, so 9 hens will eat 12+6 or 18 pounds.*"**Teaching conceptually means NOT SETTING UP A PROPORTION initially. Procedures and algorithms turn off the child's sense-making and stifle intuition and number sense.**You can fight me on this all you want, folks, but you will not win here on my blog!

**So when do we introduce proportion problems involving variables and what are some good ways to solve the original problem?? I'll allow my readers to figure that out for themselves...**

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## 3 comments:

Hi,

It is nice to drop by your blog. The explanation is fantastic and it really help people to analyse their thought better. It is so important to teach concept rather than get the students to work blindly.

Once the students master the concept with understanding in depth. The problem will seems so easy.

I had a blog that focus on word problems. In Singapore, students are fearful of word problem. My blog provide worked solution to assist students on that area. The link is at

http://excelprimaths.blogspot.sg

Thank you and hope to see more interesting post from you. Happy teaching :-)

Jackson Ng

I can always find interesting posts on your blog.

Your statement “EVERYONE LEARNS BETTER WHEN PRESENTED WITH CONCRETE NUMERICAL RELATIONSHIPS BEFORE TACKLING ABSTRACTIONS.” seems like common sense, but it is not as clear cut as it may seem. It depends how that view is implemented and in what context it is implemented. For example, for many topics it makes sense to me to teach the most basic, most general, essential aspect of a concept first, using a carefully selected task, then to show how different contexts contain the main relationship but with variations that enrich the concept. Sometimes the variations look quite different from the original context, but the general concept ties them all together. This is not an inductive approach in which a commonality is abstracted from different concrete instances. I see it as first providing the central idea that helps to organize what could otherwise be disparate information. It is consistent with what Vygotsky calls “ascending from the general to the concrete.”

The work of V.V. Davydov and his associates is based on Vygotsky’s theory. His math curriculum is very different from anyone else’s. I think you will find it interesting if you are not already familiar with it. First graders use algebraic notation before even working with numbers. A detailed review of the curriculum is given by Scmittau and Morris in this pdf. Despite the title relating to algebra, the curriculum is not so much about algebra as it is about teaching a deep understanding of the concepts so that calculations flow naturally from this understanding.

Burt,

Thank you for your detailed and thought-provoking comments. I read the first few pages of Schmittau's excellent paper.

Some thoughts...

1. I recalled Vygotsky's work and how it contrasted with Piaget's. I remembered thinking back in the day that Piaget's essentially linear model of learning fails to capitalize on young children's innate ability to compare and estimate quantities. Using algebraic symbols to represent these relationships at an early age (even before 7!) makes perfect sense to me.

2. Schmittau's interpretation and implementation of the Russian model is exceptional and I would like to see this in action in the classroom. I'm tempted to contact her.

3. That being said, I need to come back to the empirical model. In the ABSENCE of the abstract foundation of the Russian model, my recommendations are intended to help teachers working with middle and high schoolers who have not been exposed to this quantitative/abstract model It is difgicult to "unlearn" what they have been taught. However I believe it is possible to incorporate some of Vygotsky's and Davydov's ideas into instruction with older students.

This is best exemplified by the classic "There are twice as many girls as boys" problem, the post which has more views on my blog than any other. I suggested we start by asking "Are there more girls or more boys?" From there move to G>B and how do We make the quantities equal, G=2B or 2G=B? This is IMO is similar to the Russian approach.

3. My generalization of using a concrete inductive approach is based on working with older students. If I could begin from the ground up I would blend the 2 approaches. Ironically when I've worked with my own children and grandchildren from an early age I probably used Vygotsky's approach most often. It's the algebraic representation however which distinguishes his ideas and which is worth exploring further.

I hope to continue this conversation as I become better informed by Schmittau's paper.

Thanks again, Burt. We do not disagree!

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