Standards and the COLT Workbook
Diverse math methods for diverse kids
A story behind the COLT Workbook.
I was teaching 7th grade math and starting the review of negative numbers. I showed how 2-5=-3. That's when a girl in the middle of the classroom blurted out, "You can't do that!"
"What do you mean?" I replied. "I just did!"
"No. You can't subtract a big number from a small number."
"Yes you can, because now you have negative numbers."
"But," she declared, "That's impossible!"
"Why?"
"My teacher told me so."
That story encapsulates many students' attitudes when they get to the middle grades. I don't fault them for not knowing, but why are they told that negative answers are impossible or wrong or not allowed or just too difficult? With sets, that is true, but I think every student has seen a number line by grade 3. And I think everyone is familiar with negative temperatures and the concept of debt.
I think this misconception of negatives is a product of our mainstream curriculum and thinking in the early grades. Many are not ready to fully grasp the abstract concept of numbers and operations, much less the doubly abstract concept of negative numbers. Therefore, lower grade teachers sometimes say whatever they think is helpful in the short-run to get the kids through the book and out the door at the end of the year.
That is one big reason I put together the COLT Workbook with its 3-step process that handles negative numbers even if the student has never heard of them! I would prefer to have students wait until they are ready for the concept, but classroom teachers rarely have that opportunity. Even homeschoolers are still tied down to the standard sequencing.
But just because something is standard does not mean it is optimal for your student.
Let's look at the standards, but first look at the free COLT Workbook or read the "Action: Combine" chapter in the Math Actions Handbook, so you have proper background on the issue. The problem is that the standards are based on age and start way too young. Therefore, the standards-aligned textbooks teach by memory rather than understanding. This introduces misconceptions, bad habits, math anxiety, and math hatred.
Standards
What follows are the standards copied from the Common Core State Standards pdf. You can find an expanded version here. I am focused on the two arithmetic sections called, Operations and Algebraic Thinking; Number and Operations in Base Ten. The leading numbers are grade levels. My headings start with *. The key observation is to notice the incremental repetition of the same basic concepts.
* Numbers and Place Value
1--extend the counting sequence.
1--Understand place value.
2--Understand place value.
* Adding and Subtracting
1--Use place value understanding and properties of operations to add and subtract.
1--represent and solve problems involving addition and subtraction.
1--Understand and apply properties of operations and the relationship between addition and subtraction.
1--add and subtract within 20.
1--Work with addition and subtraction equations.
2--represent and solve problems involving addition and subtraction.
2--add and subtract within 20.
2--Use place value understanding and properties of operations to add and subtract.
* Multiplying and Dividing
2--Work with equal groups of objects to gain foundations for multiplication.
3--represent and solve problems involving multiplication and division.
3--Understand properties of multiplication and the relationship between multiplication and division.
3--multiply and divide within 100.
* All Four Operations
3--Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3--Use place value understanding and properties of operations to perform multi-digit arithmetic.
4--Use the four operations with whole numbers to solve problems.
4--Generalize place value understanding for multi-digit whole numbers.
4--Use place value understanding and properties of operations to perform multi-digit arithmetic.
* Miscellaneous, but basic operations
4--Gain familiarity with factors and multiples. [prep for fractions]
4--Generate and analyze patterns. [e.g. skip counting]
Big Picture
In four years of arithmetic teaching, the student journeys from beginning counting to using all four operations with big numbers. That is not much content in a groundbreaking sense. The major concepts are numbers, place value, adding with carrying, subtracting with borrowing, the times table, multiplying with regrouping, and long division. You can bury them in all the tips, tricks, and shortcuts you want, but those are the major milestones. (The MuD book on multiplying and dividing is coming soon.)
Now here is my question:
Suppose you had a class of 4th graders and they all pass a rigorous test on all those skills at the end of the year. Would it matter to you if they learned them in the previous four months versus the previous four years?
Of course, we would all be happy with both successful groups, but what do you do with the "late bloomers" for those first 3 years? I will share stories about "late" blooming brain development in the next newsletter, but the point I am driving to here is that there are only a few major concepts and we would all be happy with their mastery. Whether or not all sorts of visual, manipulative, diagrammatic or other fancy techniques were used "to promote understanding" (as Common Core would say) would not matter. There is a core package of skills we all agree on that we want kids to master.
So let's teach those skills as a package! even if we need to wait a few months or a few years to do it. (I am all for standards as a communication tool, but not as a teaching tool tied to age.)
I know, classroom teachers do not have the luxury of personal adjustments based on readiness, but homeschooling parents need to be aware of the big picture before putting their children on the mass assembly line.
However, the three-step method of combining takes both school restrictions and homeschool opportunities into account. If you can wait for understanding, great! If you can't, then here is the process to memorize that sets up the student for a lifetime of success rather than of anxiety and misconception. It also helps teachers by unifying their communication. As a 1st grade teacher moving to an 8th grade room told me, "Now I have to eat my own dog food." She was going to have to help her former students unlearn her mistaken teaching.
Organic Approach
I am not trying to get any teacher or parent to teach any math topic at a later or earlier age. I am trying to break the harmful connection between any concept and any age. True education is the presentation of a concept in a format appropriate to the able and willing learner at the time. People of any age are not raw materials to be shaped, stamped, and bent into shape. We all are plants to be nourished from the roots up. Teachers are farmers putting nutrients in the soil, if and when the plant is ready to absorb them.
My approach with Math Actions, starting with Combining, is to make the nutrients as easy as possible to swallow. The three-step process takes a daily tray full of vitamins and reduces it to an easy to remember, and easy to take, supplement. I think you will agree when you give it a try!
Teaser
The next newsletter will have many stories of learning math and reading at unexpected times and in unexpected ways, including my own daughter's! For now, I leave you with this.
"Hunter was six years old when I decided to take him off formal math studies. He didn’t do math workbooks, or math curriculum until he was 11 or 12 years old. Within a year or two, he completely caught up with all the grades that he missed." -- https://howtohomeschoolmychild.com/teaching-math-to-kids-case-studies
Am I saying you must wait until 11 or 12? No. That is what this parent happened to do. Hunter may have made the same progress at 8, 9, or 10. My point is that true education is not age based. Real learning takes place when there is both ability and desire, regardless of how much money we throw at the student.