"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Tuesday, November 3, 2015

New Ways to Learn Math

Math is now taught quite differently in elementary schools in the United States so parents are often confused if not frustrated when they look at their children's homework. There is emphasis on using different strategies. Overall, rote learning has taken a backseat while drawing or creating representations of arithmetic operations has become a staple activity.

Take for instance, dividing 50 by 4. When I was an elementary school student, I would divide 50 by 2. That would be 25 and I would then divide this answer again by 2. The final answer would then be 12 and a half. Nowadays, a student may be asked to represent this operation and one way to do so is to use fifty sticks for 50 and four circles for 4. Next, the sticks are then distributed over the four circles.


Some children get upset when forced to use multiple strategies to solve an arithmetic problem especially when they can arrive at the answer in one way they prefer and with very little work.

Dr. Wendy Bradshaw, a public school teacher in an elementary school in Florida recently resigned. She shared her resignation letter on Facebook a week ago. The post has now been shared almost 60,000 times. The letter has also been displayed on WUSF News and has received more than 300 comments. One of the comments is shown below:
What I see, as a professor who occasionally substitutes, is trying to force critical thinking instead of facts--and the history behind them--during the ages of 5-10. Don't get me wrong, critical thinking is important. However, the brain needs development and the focus must be on facts at that age. Without the facts, critical thinking through those facts to formulate hypotheses and prove those hypotheses is quite difficult. Memorization is a base level of thinking that cannot be bypassed. Yet I see it happening in the work I'm asked to have students do when I occasionally sub. I see the results of it when I do middle school (where logic should be developed). They're trying to formulate factless ideas. What ends up happening is the teacher then needs to go back to facts. It's no wonder my students in college resort to ad hominem and ad passiones arguments in their papers. A critical phase of development is completely being skipped--memorizing facts and the history behind those facts.
koine2002

Here is another one:

We use classical methods at home and know that the grammar stage from 1st-4th is essential. Logic comes next, 5th-10th roughly, then Rhetoric, and then Poetic in college. This worked for our ancestors, and is why people like Thomas Jefferson could be so eloquent in the writing of the Declaration of Independence. Ever wonder why people were so smart 100 years ago? This is why. Memorization at the sponge age, argument at the logic phase, speech making and writing at the rhetoric level, and output at the university level--poetry, drama, beauty and world view expressed so well. The modern education method will never succeed unless it goes back to classical methods.
Margaret Ryken Beaird

Bradshaw did mention in her letter that she has a doctoral degree in the field of education. She also claimed to be up to date with education research. She wrote:
...Developmentally appropriate practice is the bedrock upon which early childhood education best practices are based, and has decades of empirical support behind it. However, the new reforms not only disregard this research, they are actively forcing teachers to engage in practices which are not only ineffective but actively harmful to child development and the learning process. I am absolutely willing to back up these statements with literature from the research base, but I doubt it will be asked for. However, I must be honest....
In this regard, it may be a useful exercise to browse through a recent research paper on how to teach arithmetic to first grade students. The paper scheduled to be published in the Journal of Educational Psychology looks primarily at two approaches employed for addition and subtraction that are different from the direct approach of rote learning. The two indirect approaches are "subtraction" and "use-10". Both are discovery-based and the following are examples given in the paper:

For "subtraction", which of the following five addition items would help answer 9 - 5:
2 + 5, 4 + 5, 4 + 6, 4 +7, 5 + 6, or 5 + 7.

The "use-10" version asked, for example, which of the following five add-with-10 combinations would help answer 9 + 6:
5 + 10, 6 + 10, 8 + 10, 10 + 4, or 10 + 6.

The abstract of the paper provides conclusions that seem crystal clear:
...conceptually based indirect programs were efficacious and more successful than regular classroom instruction or the direct approach in promoting progress toward fluency and fluently itself.
One, however, simply has to look at the numbers to see if "efficacious and more successful" are really accurate adjectives. Below is part of the results of this study shown in a graph:

Practiced subtraction combinations 
(subtraction items practiced by subtraction and drill conditions)
Fluency* rate (0 to 1)
*Fluency rate for each group was the mean of its members’ proportion of fluent retrieval. A response was scored as fluent and awarded 1 point if it was correct, fast ( 3 seconds),and did not involve counting or a response bias. Otherwise, the response was scored as 0 points. Above data copied from "Does Fostering Reasoning Strategies for Relatively Difficult Basic Combinations Promote Transfer by K-3 Students?" Baroody, Arthur J.; Purpura, David J.; Eiland, Michael D.; Reid, Erin E.; Paliwal, Veena. Journal of Educational Psychology, Oct 12 , 2015, No Pagination Specified. http://dx.doi.org/10.1037/edu0000067





With regard to fluency results, the authors write down the following limitation for this study:
The fact that use-10 and subtraction interventions did not produce fluency rates of more than 32% and 17% on practiced combinations and unpracticed items, respectively, indicates that amount of practice provided by these interventions was not sufficient for many participants to automatize their targeted reasoning strategy.
These are results obtained from a highly structured classroom with high fidelity in the intervention as this has been performed by researchers yet the differences are not really that glaring between the indirect and the direct approaches.

Bradshaw speaks from her heart as she watches children cry and get frustrated as education reforms are forced into classrooms. The benefits are so small yet the harm can be so profound. Bradshaw concludes her letter with the following paragraph:
On June 8, 2015 my life changed when I gave birth to my daughter. I remember cradling her in the hospital bed on our first night together and thinking, “In five years you will be in kindergarten and will go to school with me.” That thought should have brought me joy, but instead it brought dread. I will not subject my child to this disordered system, and I can no longer in good conscience be a part of it myself. Please accept my resignation from Polk County Public Schools.

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