Putting the blame on teachers

It is easy to blame teachers, for to those who don’t look deeper, it provides the easy answer they want. But the issues which affect our education system go far deeper than the teacher. Blaming teachers disregards those things in the student’s life, such as poverty and immediate environment, which may have a pronounced effect on the person’s school performance; it also disregards anything in the education machine which may be defective. An article, dated August 9, in the New York Times, [1] begins with Sharasha Croslen who in college discovers she cannot factor a trinomial, a simple one at that. That is a sad discovery for any individual to make; but not being able to do a simple trinomial is due in main part to lack of practice. It has been said over and again that the math syllabus is unmanageably long and needs to be shortened.

When a syllabus is too long, that is, it contains too many topics, it means that the explanations have to be short, so a significant number of students will not understand what was taught. For each mathematical topic a fair deal of practice is necessary if the skill once learned, is to remain with the student a long time; additionally, some level of automaticity must be attained with certain elements such as sign rules, as this permits quick access to certain rules without much cognitive loading. Otherwise the student quickly forgets it and it is as though it was not taught to him. Forgetting means the student is unable to do succeeding topics properly; as a result very many students get cut down neither through the fault of the teacher or themselves. It means that many students get left behind as their difficulties accumulate, leading to a dislike of mathematics and a tuning out in class. The cost to those who are not stellar in mathematics is tremendous and to the country since too many people are lost along the way. It is tragic since it means that people are hired from other countries, and one would have to wonder if those who are hired are so much better in potential than those who have failed here or is it just the result of a different approach, including a shortened math curriculum.

Let us then look at a powerful statement by Mrs. Ravitch, in the document ‘Hard Work and High Expectations’ and we quote:

Notably muted in the debate has been discussion of the engagement and motivation of the students themselves. It is a curious omission, for even if we raise standards and succeed at restructuring our schools and improving the quality of our teachers, the result may be little or no improvement unless our children also increase the level of their effort. After all, now as before, it is the students who must learn more, and it is they who must do the work. [2]

The last sentence above states that it is the students who must do the work, and if this statement is said like this, ‘it is they who must do the work, if learning is to take place,’ we will regard this as an axiom or learning; it is the basis of our argument here. Whatever that work is, mathematics, calculus, trigonometry, physics, history, it is they who will do it if they are to learn. If we now return to the situation we had started with in the first paragraph, and use the quote above, we can conclude that Sharasha, and people in situations similar to hers, did not do, in high school, the work necessary to allow her to factor the trinomial she was given. We are not saying that Sharasha was given the work and did not attend to it. We are saying that the practice necessary to form a long-term retention of the skill was not given to her. So if we go back to Mrs. Ravitch we find that the axiom holds; the student, if learning is to take place must do the work. In this case, it seems that Sharasha was not given the necessary work so she could not do it. This is entirely the fault of the education system, for it does not allow teachers to deviate much even when they would want to. According to Mary Ann Whiteker a Texas School Superintendent; ‘Teachers are mandated by law to teach the state curriculum standards.’ [3]

Here is a statement from an online article by Ms. English, ‘School curriculum proceeds rapidly because teachers follow pacing guides to ensure they’re on track for mandatory state tests.’ [4] This is something which has been mentioned over and again by different people. Mrs. Linda Darling-Hammond says the same thing using different words: ‘high-achieving nations teach about half as many topics each year as American schools do, treating them more deeply, with greater opportunity to work on a range of solution strategies and to engage students in applying what they are learning.’ [5] Why policy makers and bureaucrats continue to ignore such an important recommendation is difficult to understand; a lessening of the repeated topics is not difficult to do. Some topics such as the circle theorems in geometry could easily be left out. Let us say a topic such as Pythagoras’s Theorem is covered in eight grade in such a way that it does not have to be covered again, then when those student need to use it in a higher grade they will know, really know, how to use it and the teacher can move on and attend to the topics which use it and not waste time; presently that is not the case and there is too much reteaching to do. When the teacher asks for Pythagoras’s theorem in ninth grade we must see all hands going up with a resounding ‘Sir please call me I got this.’ Yes we know it was taught and not passed over quickly.

Lessening the number of topics taught in mathematics or reducing repetition will prove to be beneficial to student learning. It would permit for better and slower development of each topic, allowing teachers to give the kind of practice which embeds skills in the students’ minds. This is not a matter of professional development or teacher failure; if enough time is not given the topic cannot be adequately covered, and a tragic number of students will not get it. The four operations of algebra one are vital to the rest of mathematics and those who do not get it will experience difficulties with mathematics for some time, possibly putting it down as soon as they can. It is there that many students begin to lose their way. Any attempt to hurry will result in painful confusion, children lining up with questions; they will not get it. While addition is simple it has to be massaged in, otherwise when they reach subtraction they begin to confuse the rules of addition and subtraction. Indeed the decision as when to move on should be left to the teacher and not to some ‘pacing guide;’ continuing to keep from the teacher the decision as when to move on to another topic shows on the part of policy makers a complete disregard for how people learn and an uncaring attitude for students. But the topic is not well done if the student does not get a fair amount of practice, otherwise the skill will soon be lost. It is more than likely that Sharasha was taught trinomial factoring but the practice which was necessary for embedding the skills was not done and so she, and a number of people like herself end up not getting a proper grasp of relatively simple mathematics. A fair grasp includes not just understanding the material but enough practice that the skill remains for a long time. Her Asian counterpart would have had much more practice and so much more able to remember what was taught to her. If a child gets lost in algebra one, his or her mathematical life is in danger and so these steps need to be taken with care, proceeding slowly. Each step must be done with enough time that automaticity, where necessary, is achieved otherwise when the child moves from say addition to subtraction the young mind gets confused and this is difficult to undo, because some of them become frightened and insecure. The teacher will not move on until the lesson’s objectives are learned and properly. When the pacing is determined by a rule it is there that the failure occurs and the student becomes confused, frustrated and gives up eventually leading to a rejection of mathematics.

Let us return to Ms. English, an American Teacher, who states the problem like this:

The American education system has become a form of, “survival of the fittest,” and this system does not serve *all children* well. Students who read well, learn quickly, and reproduce information without notes can be successful, but for every child who is successful, there may be five or more who struggle. A student who is two weeks behind in mastering math concepts, for example, will continue to fall behind as the teacher “keeps the pace” to cover new concepts, and there is little time for teachers to re-teach the material. [6]

This is uttered by a practicing teacher and like the rest of teachers her heart breaks when she has to move on, because of an utterly ruthless pacing guide, knowing that too many students do not get it. For every student who gets through there are five or more who struggle. Uncaring about students the pacing guide demands the teacher goes on to complete the syllabus, leaving failing students along the way, having to face the frustration of mathematics for the rest of the year or for years. If a system fails so many student then it is time to make some meaningful change. Our students are in school with a mathematics syllabus which, as our teacher states, produces five struggling students for every one which passes. Our students have little choice but to tune out when they face such daunting difficulties. According to William Schmidt in 2005: “In other countries, they might spend a month on a topic while we spend days on a topic.” [7]Anyone they turn to, parent, teacher, or friend shares the same experience and similar assumptions that this way of teaching mathematics is the only way. They turn away eventually from mathematics.

It is not that policy makers are unaware of the problem but they have adopted approaches that largely unworkable and in trying to address the problems treated the teachers with singular disrespect; something which would ensure that their goals would fail. In order to get students to pass the standardized tests teachers are put under crushing pressures, consequently focusing less on substance and more on tests grades. In the first place there is only so much a teacher can do, but additionally the principle of asking the teacher to get students to pass would run counter to many of the teachers’ principles, for most of us think that students should work or be taught how to work. If the teacher has to get students to pass tests, then the student becomes more dependent on the teacher.

#### Not working

In the article by Jennifer Medina, [8] she quotes John Garvey, the liaison between CUNY, the City University of New York, and the public schools, as saying in reference to a student, “This is a student who thinks he or she has been doing pretty well, but their first experience is being told you are not good enough,” he said. “All of their confidence and determination is being undermined.” Such a statement presents us with an immense problem. Students, having spent many years in high school, having graduated, are now being told that their work is not good enough. So the poor quality work they were turning in to their high school teacher was being accepted with praise? Again we see examples of student not doing the work, but in this case the high school teachers are telling them that they were doing acceptable work.

Then you say well why were the teachers not giving them the grade they deserved. The answer to that question was given by a teacher J. Maher in a letter to the editor in response to a parent, who wanted to know why they promoted students who had not done satisfactorily. He said that they promoted in order to keep their jobs. Parents were not doing their part and teachers could not do the job by themselves. We come again to the question of work. If the student is given the grade they deserve, teachers will be dismissed. It is a difficult issue but blaming teachers is not a logical answer.

These kids above are poor kids but the better kids are also affected by the lack of work. According to Jeff Guo, a University of Toronto study was done and its aim was to understand why students who did well in high school did poorly in college. According to Jeff many of the kids who did well in high school did so without the exertion of effort and got their A’s without hard work. [9] When they encountered difficulty in college they opted out of the engineering or some other math requiring class. So just like the poorer kids these kids had not been challenged and they were told, with more justification than the New York kids, that they were doing well. They had not been stretched.

Again we see how the ‘mile wide and inch deep’ curriculum does an injustice to our students. The pacing guide does not allow the teacher to go into depth which is where the stronger students would encounter challenge with more difficult problems. Since these require harder work the better students are also not stretched and they are able to pass without doing really challenging work. Oh yes the bright and gifted will always get through but they are not being challenged.

We see the wisdom of the Finnish when they gave the teachers a great degree of autonomy to make adjustments to curriculum so they are able to decide when to leave a topic, which is when their students understand the topic. Such autonomy would be welcome by American teachers as it would allow them to take more time in teaching topics and move on only when students understand the material as determined by teacher given tests. The situations within schools and classrooms are complex and dynamic so allowing for more teacher autonomy would allow them to adjust as necessary to reach their teaching goals. If teachers are allowed to implement changes through agreement they would also be committed to the decisions which come out of their meetings.

We have been asking teachers to work, afraid it would seem, to ask students to do the work, afraid that there would be failure and parents would cry out to politicians. Yet, unless someone can come up with a different principle, student effort will continue to be required for them to perform well, and it is for us, the adults at present, to establish the rules which will give us in the future well developed people at the helm of this country.

#### What is work then?

What is meant by the word work when we speak here. In relation to mathematics, let us say algebra one, the work would consist of abundant practice at the early stage of a topic and then leading to the challenging problems in each topic. Difficult problems take time, class discussion and cannot be hurried any more than the simple part of the topic. At each stage the teacher assesses to see if the student has acquired the skill and moves on when the skill is mastered.

Practice in mathematics is recognized as important and Barbara Oakley states this in her book, ‘A Mind for Numbers.’ Says she, ‘Solving a lot of math problems provides an opportunity to learn why the procedure works the way it does or why it works at all.’ [10] She also states the need for difficult problem and says. Difficult problems, the ones which takes long to solve, produce befuddlement. Says she:

Befuddlement is a healthy part of the learning process. When students approach a problem and don’t know how to do it, they’ll often decide they’re no good at the subject. Brighter students, in particular, can have difficulty in this way—their breezing through school leaves them no reason to think that being confused is normal and necessary. [11]

Ms. Oakley then, recognizes that difficulty is a normal part of problem solving. When we say the student did not do the work, this is a part of what we are referring to. Learning is solidified and taken to a higher level when students encounter and do difficult problems. Note she mentions that these problems take away some of the swag of the bright kids who now along with those who are not so good have to struggle. These problems play an important role in mathematics and any attempt to avoid them will affect profoundly the mathematical development of the student. Many of these problems will lead to solutions which do not come out when working on the problem but when the person is more relaxed.

Says Mason in ‘Thinking Mathematically,’ ‘Probably the most important lesson to be learned is that being stuck is an honorable state and an essential part of improving thinking.’ There is then, agreement that working on difficult problems which offer quite great deal of learning, is necessary. These problems take thinking, understanding, strategizing to a higher level and without them the education is not complete; they award students with a better grasp of whatever topic is being covered and failure to present the students with these will be to deprive them of facing difficulty. We are saying that Sharasha did not get any of these, yet she was promoted; it is natural that she would find it impossible to solve that problem, a simple trinomial since the coefficient of is one; the expression, is of a higher order and comes when they have finished simpler ones. Solving a few of the harder ones requires more making it easier to remember trinomials. But covering difficult trinomials takes time also and cannot be rushed.

Britain has taken it seriously and is importing mathematics texts books and teachers from China to rejuvenate her math program. There needs to be change in how the curriculum is covered and teachers need to be given a much bigger part in how to implement it. It is not about copying from any other nation but if there are principles which are worth adopting then that should be done.

[1] Medina, Jennifer (2010): Schools Are Given a Grade on How Graduates Do. In *New York Times*, 8/9/2010. Available online at http://www.nytimes.com/2010/08/10/education/10remedial.html?pagewanted=all&_r=0, checked on 7/18/2013.

[2] Hard work and High Expectations, p 1.

[3] http://lufkindailynews.com/news/community/article_76ca4c86-3873-11e4-8dd8-0019bb2963f4.html

[4] English, Janet (2013): US Teacher Gets Finnish Lesson in Optimizing Student Potential. Part 1. Available online at http://oecdinsights.org/2013/09/23/us-teacher-gets-finnish-lesson-in-optimizing-student-potential/, checked on 2/1/2014.

[5] Darling-Hammond, Linda (2010): The flat world and education. How America’s commitment to equity will determine our future. New York: Teachers College Press.

[6] [6] English, Janet (2013): US Teacher Gets Finnish Lesson in Optimizing Student Potential. Part 1. Available online at http://oecdinsights.org/2013/09/23/us-teacher-gets-finnish-lesson-in-optimizing-student-potential/, checked on 2/1/2014.

[7] http://chicagoreporter.com/math-teaching-in-us-inch-deep-mile-wide/

[8] Medina, Jennifer (2010): Schools Are Given a Grade on How Graduates Do. In *New York Times*, 8/9/2010. Available online at http://www.nytimes.com/2010/08/10/education/10remedial.html?pagewanted=all&_r=0, checked on 7/18/2013.

[9] Guo, Jeff. „Why students who do well in high school bomb in college.“ *Washington Post*, 21.09.2016. Zuletzt geprüft am 12.08.2017. https://www.washingtonpost.com/news/wonk/wp/2016/09/21/why-students-who-do-well-in-high-school-bomb-in-college/?utm_term=.e97fc43bce0c.

[10] From A Mind for Numbers by Barbara Oakley, p 59.

[11] Barbara Oakley p 22

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